The rule used to be that if you were a recruited athlete--you had offers from other D-I schools (and maybe lower division also)--you had to be put on an athletic scholarship.
That was one of the things that got the SU lax program in trouble in the late '80's-early '90's, when one of the star players was paid out of other funds because his mother worked for the university. That freed up money to recruit other athletes to the program.
On the other hand, I'm not sure what the story was with Doug McDermott of Creighton hoops fame. He played for his father, and for some reason, I think he may have been counted as a walk-on his last year, because they had run out of scholarships. Although it may have been someone else on the roster who gave up his athletic scholarship.