Founded in 1636 and chartered in 1650, Harvard calls itself "the oldest institution of higher learning in the United States," and there is no challenge. William and Mary calls itself "the second oldest college in America, and the first to become a university," acknowledging Harvard's claim.
Although no one challenges Harvard's claim on being the oldest higher learning institution in the United States, an argument about the actual founding date could be made. Harvard uses the earliest possible date, 1636, when it was first chartered by the Massachusetts Bay Colony. It has operated under the same corporation, the President and Fellows of Harvard College, since 1650.