JOEL - It does happen, although not too often; most are from the lineage of Ephraim. Isaac received the promise that through his lineage the promises given Abraham would be transferred to his posterity. Jacob in turn received the same promise and therefore blessed his sons. Jacob's eldest son, Reuben, lost the birthright because of sin, and it was given instead to Joseph and then passed on to Abraham's descendants through Joseph's son Ephraim. Because of this birthright the gathering of Israel that takes place in our day is a gathering of Ephraim first and then of the other tribes. Most members of the Church, therefore, are of Ephraim, however a person's lineage can come through any of the sons of Israel.
Elder Russell M. Nelson said:

"Are you Jewish? That precious lineage may be claimed if your ancestors are from the loins of Judah. But most of us are of the lineage of Joseph through Ephraim or Manasseh. Joseph's was the lineage selected to pioneer the gathering of Israel, the seed to lead throughout the world in blessing all the nations of the earth."
(Perfection Pending, and Other Favorite Discourses, Russell M. Nelson, p. 207)

President President James E. Faust said the following:

"As Joseph Fielding Smith stated, “The great majority of those who become members of the Church are literal descendants of Abraham through Ephraim, son of Joseph.”. However, Manasseh, the other son of Joseph, as well as the other sons of Jacob, have many descendants in the Church. There may be some come into the Church in our day who are not of Jacob’s blood lineage. No one need assume that he or she will be denied any blessing by reason of not being of the blood lineage of Israel. The Lord told Abraham, “And I will bless them through thy name; for as many as receive this Gospel shall be called after thy name, and shall be accounted thy seed, and shall rise up and bless thee, as their father.” (Abr. 2:10)
(James E. Faust, “Priesthood Blessings,” Ensign, Nov. 1995, 62)

"Since families are of mixed lineage, it occasionally happens that members of the same family have blessings declaring them to be of different lineage. There has been an intermixture of the tribes one with another. One child may be of Ephraim, another in the same family of Manasseh, Judah, or one of the other tribes. The blood of one tribe, therefore, may be dominant in one child and the blood of another tribe dominant in another child, so children from the same parents could belong to different tribes."
(James E. Faust, “Your Patriarchal Blessing,” New Era, Nov. 2005, 4)

Patriarchs have been counseled:
"A vital part of every patriarchal blessing is the declaration of lineage. . . . The patriarch should be responsive to the whisperings of the Spirit as he identifies lineage and the special promises and blessings attendant thereto. . . . The declaration of lineage is to come by the promptings of the Holy Ghost. This inspiration can come to the patriarch regardless of the race or nationality of the person receiving the blessing."
(Information and Suggestions for Patriarchs (Salt Lake City: The Church of Jesus Christ of Latter-day Saints)

So our lineage could be identified with any one of the sons of Israel, and according to President Faust members of the same family can be from a different lineage.
I once knew someone who had received a patriarchal blessing and in it learned that he was from the lineage of Dan. Years later he lost his copy of the blessing and it could not be found in the church records. So he went to his current patriarch and received a new blessing, without telling the patriarch his earlier pronounced lineage. The new Patriarch was inspired to pronounce the same lineage in the new blessing.