JOEL - In the Book of Mormon we read:
"O how great the holiness of our God! For he knoweth all things, and there is not anything save he knows it." (2 Nephi. 9:20, See also Alma 26:35)

I think we can say with certainty that God knows everything that there is to know; that He knows the history of the universe and mankind from the beginning to the end. He does not need to increase in any attribute, including knowledge, because He represents the fulness of every attribute a god can have.

President Joseph Fielding Smith said:
"Do we believe that God has all "wisdom"? If so, in that, he is absolute. If there is something he does not know, then he is not absolute in "wisdom," and to think such a thing is absurd. Does he have all "power"? If so then there is nothing in which he lacks. If he is lacking in "wisdom" and in "power" then he is not supreme and there must be something greater than he is, and this is absurd. Where has the Lord ever revealed to us that he is lacking in knowledge? That he is still learning new truth; discovering new laws that are unknown to him? I think this kind of doctrine is very dangerous. I don't know where the Lord has ever declared such a thing. It is not contained in any revelation that I have read. Man's opinion unaided by the revelations of the Lord, does not make it so." (Joseph Fielding Smith, Doctrines of Salvation)

The only thing God increases in is the glory that is added to him as His children multiply, grow and progress in perfection as explained by the Prophet Joseph Smith:

"What did Jesus do? Why; I do the things I saw my Father do when worlds come rolling into existence. My Father worked out his kingdom with fear and trembling, and I must do the same [that is Christ must do the same]; and when I get my kingdom. I shall present it to my Father, so that he may obtain kingdom upon kingdom, and it will exalt him in glory. He will then take a higher exaltation, and I [Christ] will take his place, and thereby become exalted myself. So that Jesus treads in the tracks of his Father, and inherits what God did before; and God is thus glorified and exalted in the salvation and exaltation of all his children." (Teachings of the Prophet Joseph Smith)

I am not sure where you are hearing anything other than this but, any attempts to suggest that God is in some way increasing in anything else is just man's attempt to make sense out of something that our mortal finite minds cannot understand.

SARAH - Welll, I first got the idea that there was a divide because in the encylopedia of mormonism where as far as omniscience they state:
"Latter-day Saints differ among themselves in their understanding of the nature of God's knowledge. Some have thought that God increases endlessly in knowledge as well as in glory and dominion. Others hold to the more traditional view the God's knowledge, including foreknowledge of future contingencies, is complete."

Wilford Woodrff recorded this is his journal on 17th September, 1854:
"President Young said that the doctrine taught in the Seer (an article Orson Pratt had written) that God had arrived at that state whereby He could no advance any further in knowledge, power, glory was a false doctrine and not true. That there never will be a time to all eternity when all the Gods of Eternity will cease advancing in power, knowledge, experience and glory, for if this were the case, eternity would cease to be and the glory oF God would come to and end. But all the celestial beings will continue to advance in knowledge and power, worlds without end. (empahsis added).
...He told Brother Pratt to lay aside his philosophical reasoning and get revelation from God to govern him and enlighten his mind more. And it would be a great blessing to him to lay aside his books and go into the canyons as some of the rest of us were doing and it would be better for him. He said philosophy injured him in a measure.

President Wilford Woodruff also said the "God himself in increasing and progressing in knowledge, power, dominion, and will do so worlds without end."

Lorenzo Snow said "We will continue on improving, advancing, and increasing in wisdom, intelligence, power and dominion, worlds without end."

JOEL - Those are some interesting quotes, and I do know that there have been debates on this subject in the past. If we are to take these statements at face value, it is possible that there are two ways to look at eternal progression. There is the eternal progression of man where we are striving to become like God, and from our perspective, God is all-knowing and all-powerful and does not progress in the same way as we do.
Then there is the eternal progression of God, where in His own sphere of existance, He is, in some way, progressing in knowledge and power in a way we cannot comprehend.

There is a scripture in the Doctrine and Covenants which says:
"All truth is independent in that sphere in which God has placed it, to act for itself, as all intelligence also; otherwise there is no existence" (D&C 93: 30)

In other words, there is a different deffinition of truth or what constitutes truth in our sphere of existance compared to that in God's sphere of existance. In our sphere of existance and as far as we are concerned, the truth is that God is all-knowing and all-powerful and does not continue to learn; but that is not the case in His sphere of existance. But this is of course all speculation.
Like I said before, I think the ability to completely understand the nature of God is beyond the ability of our finite mortal minds. It is possible that not even our past prophets were in agreement, which seems to indicate that God has not specifically revealed this information to us yet; or at least not to the degree that we can call it official Church doctrine. We only have the opinions of man trying to make sense of how we(or a god) can eternally progress and yet be omniscient and omnipotent at the same time. For the moment I think we should just worry about our own sphere of existance and stick with what the scriptures say and leave it at that.

"Now have we not reason to rejoice? Yea, I say unto you, there never were men that had so great reason to rejoice as we, since the world began; yea, and my joy is carried away, even unto boasting in my God; for he has all power, all wisdom, and all understanding; he comprehendeth all things, and he is a merciful Being, even unto salvation, to those who will repent and believe on his name." (Alma 26:35)

"O how great the holiness of our God! For he knoweth call things, and there is not anything save he knows it." (2 Nephi. 9:20)

"He comprehendeth all things, and all things are before him, and all things are round about him; and he is above all things, and in all things, and is through all things, and is round about all things; and all things are by him, and of him, even God, forever and ever." (D&C 88:41)