Sort of right.

A player could not play one season (had to not play at all, back then. now its 4- games) and a coach could have still only given that player a scholarship/spot on the team for 4 years. They don't, necessarily, HAVE TO let a guy play 5 years. But you still had eligibility (ability to play per NCAA). And before the transfer rules changed, you really couldn't go find a second team, as easily. So the coach COULD essentially deny you a redshirt year by saying "you've been here 4 years, thanks for playing."