A lot of the classes are to ensure the students get a proper education. Someone in their early twenties would presumably already have attended another school, and therefore would not be required to take any of the standard classes.

Adults can be brought in to stay at the Institute whilst they discover their powers, but they're more likely to receive personal tutoring than be placed in any of the classes with the teenagers - generally mutants' powers become active during puberty (or earlier, if exposed to trauma or the like), so there aren't too many 'new' mutants over the age of eighteen.