Can an individual progress from one kingdom to another?

President George Albert Smith - 
There are some people who have supposed that if we are quickened telestial bodies that eventually, throughout the ages of eternity, we will continue to progress until we will find our place in the celestial kingdom, but the scriptures and revelations of God have said that those who are quickened telestial bodies cannot come where God and Christ dwell, worlds without end. (Conference Report, October 1945, p.172)

Resurection - status of an individual.

Alma 11:42–45

Alma 34:32-34

2 Nephi 28:7-10

D&C 137:1–10

D&C 131:1–4

Revelation 20:123 Nephi 27:14

John 14:2

1 Corinthians 15:40–42, including footnote 40a;

 D&C 88:20–39130:18–19

George Albert Smith
There are some people who have supposed that if we are quickened telestial bodies that eventually, throughout the ages of eternity, we will continue to progress until we will find our place in the celestial kingdom, but the scriptures and revelations of God have said that those who are quickened telestial bodies cannot come where God and Christ dwell, worlds without end. (Conference Report, October 1945, p.172)

Spencer W. Kimball
After a person has been assigned to his place in the kingdom, either in the telestial, the terrestrial, or the celestial, or to his exaltation, he will never advance from his assigned glory to another glory. That is eternal! That is why we must make our decisions early in life and why it is imperative that such decisions be right. (The Teachings of Spencer W. Kimball, p.50; The Miracle of Forgiveness, p.243-244)

Joseph Fielding Smith
It has been asked if it is possible for one who inherits the telestial glory to advance in time to the celestial glory?

The answer to this question is, No!

The scriptures are clear on this point. Speaking of those who go to the telestial kingdom, the revelation says: "And they shall be servants of the Most High; but where God and Christ dwell they cannot come, worlds without end."

Notwithstanding this statement, those who do not comprehend the word of the Lord argue that while this is true, that they cannot go where God is "worlds without end," yet in time they will get where God was, but he will have gone on to other heights.

This is false reasoning, illogical, and creates mischief in making people think they may procrastinate their repentance, but in course of time they will reach exaltation in celestial glory.

Now let us see how faulty this reasoning is. If in time those who enter the telestial glory may progress till they reach the stage in which the celestial is in now -- then they are in celestial glory, are they not, even if the celestial has advanced? That being the case (I state this for the argument only, for it is not true), then they partake of all the blessings which are now celestial. That means that they become gods, have exaltation, gain the fulness of the Father, and receive a continuation of the "seeds forever." The Lord, however, has said that these blessings, which are celestial blessings, they may never have; they are barred forever!

The celestial and terrestrial and telestial glories, I have heard compared to the wheels on a train. The second and third may, and will, reach the place where the first was, but the first will have moved on and will still be just the same distance in advance of them. This illustration is not true! The wheels do not run on the same track, and do not go in the same direction. The terrestrial and the telestial are limited in their powers of advancement, worlds without end. (Doctrines of Salvation, 2:31-32)

James E. Talmage
The Lord has told us of places prepared for those entitled to salvation. He has told us that those who will keep all the laws and commandments of God can come where he is and shall be heirs of celestial glory and power. And he has told us of lesser degrees unto which others who have failed to rise to the occasion of laying hold on the blessing of eternal life, in its fulness, shall come; and concerning the last of these kingdoms of glories, known to us as the Telestial, the Lord has said that it excels all that the human mind can conceive in glory, and yet the one in the Telestial Kingdom is condemned so far as his actions have rendered him incapable of attaining to the higher glories and blessings which mean power and advancement. (Conference Report, April 1930, p.96)

Bruce R. McConkie

There are those who say that there is progression from one kingdom to another in the eternal worlds or that lower kingdoms eventually progress to where higher kingdoms once were.

This belief lulls men into a state of carnal security. It causes them to say, "God is so merciful; surely he will save us all eventually; if we do not gain the celestial kingdom now, eventually we will; so why worry?"

It lets people live a life of sin here and now with the hope that they will be saved eventually.

The true doctrine is that all men will be resurrected, but they will come forth in the resurrection with different kinds of bodies - some celestial, others terrestrial, others telestial, and some with bodies incapable of standing any degree of glory. The body we receive in the resurrection determines the glory we receive in the kingdoms that are prepared.

Of those in the telestial world it is written:

"And they shall be servants of the Most High, but where God and Christ dwell they cannot come, worlds without end" (D&C 76:112).

Of those who had the opportunity to enter into the new and everlasting covenant of marriage in this life and who did not do it the revelation says:.

Therefore, when they are out of the world they neither marry nor are given in marriage; but are appointed angels in heaven; which angels are ministering servants, to minister for those who are worthy of a far more, and an exceeding, and an eternal weight of glory.

For these angels did not abide my law; therefore, they cannot be enlarged, but remain separately and singly, without exaltation, in their saved condition, to all etemity; and from henceforth are not gods, but are angels of God forever and ever. [D&C 132:16-17]

They neither progress from one kingdom to another, nor does a lower kingdom ever get where a higher kingdom once was. Whatever eternal progression there is, it is within a sphere. ("The Seven Deadly Heresies," Classic Speeches, Provo, UT: Brigham Young University, 1994, pp. 175-176)

Franklin D. Richards
Hiram [Smith] said Aug 1st [18]43 Those of the Terrestrial Glory either advance to the Celestial or recede to the Telestial [or] else the moon could not be a type [viz. a symbol of that kingdom]. [for] it [the moon] "waxes & wanes." (From a sermon transcribed by Franklin D. Richards in Words of the Prophet Joseph Smith, pg. 24, 1 August 1843)

Wilford Woodruff
In conversing upon various principles President [Brigham] Young said none would inherit this Earth when it became celestial and translated into the presence of God but those who would be crowned as Gods and able to endure the fullness of the presence of God, except they would be permitted to take with them some servants for whom they would be held responsible. All others would have to inherit another kingdom, even that kingdom agreeing with the law which they had kept. He said they would eventually have the privilege of proving themselves worthy and advancing to a celestial kingdom, but it would be a slow progress. (Journal of Wilford Woodruff, 5 Aug 1855)

Franklin D. Richards
The Savior tells us that the terrestrial glory, or kingdom, is likened unto the glory of the moon, which is not of the brightness of the sun, neither of the smallness nor dimness of the stars. But those others who have no part in marrying or giving of marriage in the last resurrection, they become as stars, and even differ from each other in glory; but those in the terrestrial kingdom are those who will come forth at the time when Enoch comes back, when the Savior comes again to dwell upon the earth; when Father Abraham will be there with the Urim and Thummim to look after every son and daughter of his race; to make known all things that are needed to be known, and with them enter into their promised inheritance. Thus the people of God will go forward. They will go forward, like unto the new moon, increasing in knowledge and brightness and glory, until they come to a fullness of celestial glory. (Journal of Discourses Vol. 25:236, 17 May 1884)

Joseph F. Smith
Once a person enters these glories there will be eternal progress in the line of each of these particular glories, but the privilege of passing from one to another (though this may be possible for especially gifted and faithful characters) is not provided for. (Improvement Era 14:87, November 1910)

J. Reuben Clark, Jr
I am not a strict constructionalist, believing that we seal our eternal progress by what we do here. It is my belief that God will save all of His children that he can: and while, if we live unrighteously here, we shall not go to the other side in the same status, so to speak, as those who lived righteously; nevertheless, the unrighteous will have their chance, and in the eons of the eternities that are to follow, they, too, may climb to the destinies to which they who are righteous and serve God, have climbed to those eternities that are 

B. H. Roberts

These are the great divisions of glory in the world to come, but there are subdivisions or degrees.  Of the telestial glory it is written: “And the glory of the telestial is one, even as the glory of the stars is one, for as one star differs from another star in glory, even so differs one from another in glory in the telestial world” [D&C 76:98].  From this it is evident that there are different degrees of glory within the celestial and the telestial glories; and though we have no direct authority for the statement, it seems but reasonable to conclude that there are different degrees of glory in the terrestrial world also.  It appears but rational that it should be so, since the degrees of worthiness in men are almost infinite in their variety; and as every man is to be judged according to his works, it will require a corresponding infinity of degrees in glory to mete out to every man that reward of which he is worthy, and that also which his intelligence will enable him to enjoy.

The question of advancement within the great divisions of glory celestial, terrestrial, and telestial; as also the question of advancement from one sphere of glory to another remains to be considered. In the revelation from which we have summarized what has been written here, in respect to the different degrees of glory, it is said that those of the terrestrial glory will be ministered unto by those of the celestial; and those of the telestial will be ministered unto by those of the terrestrial–that is, those of the higher glory minister to those of a lesser glory.  I can conceive of no reason for all this administration of the higher to the lower, unless it be for the purpose of advancing our Father’s children along the lines of eternal progression.  Whether or not in the great future, full of so many possibilities now hidden from us, they of the lesser glories after education and advancement within those spheres may at last emerge from them and make their way to the higher degrees of glory until at last they attain to the highest, is not revealed in the revelations of God, and any statement made on the subject must partake more or less of the nature of conjecture.  But if it be granted that such a thing is possible, they who at the first entered into the celestial glory–having before them the privilege also of eternal progress–have been moving onward, so that the relative distance between them and those who have fought their way up from the lesser glories may be as great when the latter have come into the degrees of celestial glory in which the righteous at first stood, as it was at the commencement.  Thus:  Those whose faith and works are such only as to enable them to inherit a telestial glory, may arrive at last where those whose works in this life were such as to enable them to entrance into the celestial kingdom–they may arrive where these were, but never where they are. (New Witnesses for God, 1:391-392)

John A Widtsoe:
"There is also a difference in possible achievement in the different glories. Granite cannot be carved with wooden tools. So it is in the glories of the hereafter. Those of the celestial kingdom have so lived as to achieve Godhood itself. Those of an inferior glory cannot reach that far. The deeds on earth become tools of achievement in the heavens." 
(Evidences and Reconciliations)

Interesting quotes:
In the temple we progress through the telestial and terrestrial kingdoms in order to be saved in the celestial.

The flaw is in the application of 'Where God and Christ dwell they cannot come, worlds without end' [D&C 76:112].

It is based on the also erroneous assumption that those in the Celestial never move beyond (a statement D&C 130:10 shows to be false).

Yes, those who are resurrected to a Telestial or Terrestrial level can NEVER go where Christ and God are.  But if Christ, God and the Celestialized children progress onwards and upwards beyond Celestial to higher kingdoms, this just means the Telestial and Terrestrial beings will never catch up to their presence.  It does NOT prevent a Terrestrial Kingdom and its inhabitants from progressing and perfecting themselves to a Celestial level.

As long as beings exist they are either progressing or regressing.  God's work and glory is to make his children into beings like himself.  That doesn't stop on the other side of the veil.  Those who fail to live up to Celestial law here may have a path to follow that we don't yet realize.  But they may individually or collectively as a kingdom continue to progress until worthy of Celestial glory, even if Heavenly Father and Christ's kingdoms by then have moved upward.

What would be the point in the Celestial ministering in the Terrestrial if not to help them to come up to that level?

It makes no sense to put them in a finite box (no matter the size) any more than it does to put God in a finite box and say God is done progressing.

I believe in eternal progression - for every being that is granted an eternal existence.  The alternative is too nonsensical and horrendous to even consider.  (There was a wonderful episode of Star Trek that proved this perfectly).

Logic more than scripture, since there is very little revealed about our post-resurrection progression.

I liked Rob's example of moving through the temple as well.

To say that someone can only progress within an assigned kingdom is to say: 

This Terrestrial kingdom contains 100 gallons of truth and knowledge (whereas the Celestial might contain a million gallons).  I can progress, gaining a gallon worth of truth and light every millennia of my existence.  Or maybe I'm slow and only gain 1/2 a gallon every millennia.  No matter how slow my progression is, if I am limited to any space/kingdom/container I will eventually achieve EVERYTHING there is to do at that level.  It could take billions of years, but eventually a progressing terrestrial being will have perfected themselves terrestrially AND done absolutely everything there is to do in a terrestial glory.  I am an eternal being with nothing to do for eternity.

Then I'm supposed to believe there is nothing else for them to do for the rest of their eternal existence.  I cannot accept that.

I do not know how those in the Terrestrial or Telestial would ever achieve exaltation (they would need to marry for instance) or what path their progression will take but I am not claiming any special knowledge on the how.  All I know for certain is that God did not create children with eternal existence to live in a finite kingdom.  If we are eternal, then there is a reason to exist eternally.

If we are to have an end to what we can achieve because bounds have been set, then there is no point in being eternal - we only need to exist long enough to achieve the limits of our location.

The best scriptures on the subject only hint at it:

D&C 130:10 tells us when we enter the Celestial Kingdom there are higher kingdoms to aim for.

Abraham 3:18-19 tells us that of any two intelligences one always has greater light - in a finite body of knowledge (terrestrial, celestial) eventually all would learn the same amount of light and be equal.

The temple language is specific that we do in fact go from one kingdom, learning line upon line, precept upon precept, until we get to where we are prepared to enter into the next higher kingdom. It is the least understood doctrines in our church. The BoM prophets were in large part speaking on behalf of their own beliefs and not what is really truth. Latter day revelation teaches us that labors are performed in the spirit world and that spirits there do work to perfect themselves according to the laws and ordinances of the temple. This work is open to both those who never had the opportunity and those who died in their sins who rejected the prophets.

I do believe everyone will still be able to progress in glory if they really want to, though, and for as long as others are trying to teach them. Glory is all about intelligence, so to progress in glory involves gaining more intelligence and if a person is already as intelligent as he or she knows how to be, with as much intelligence as he or she has, then someone else with more intelligence is needed to give them more intelligence or intelligent information, otherwise the less intelligent person will be stuck with no way to gain any more intelligence than he or she already has. I would like to think that my visits to try to teach less intelligent people would do them some good, and my door will always be open to anyone who can help me gain more intelligence than I have at any given moment in eternity.