So, "cut" is a relative word in my opinion. What my son described to me sounds more like "self-imposed attrition with the coaches help" but that takes too long to say.
Yes. My son's recruiting class had two players "cut". One (position player) realized quickly freshmen year he wasn't going to play, and left to start his own internet business which is thriving. He was a multi-millionaire by the time he graduated. The other player (RHP pitcher) wasn't so quick to realize that he wasn't going to play, and the coach reminded him of that fact in the Fall of sophomore year. He left the team at that point. Both guys had something better to do with their lives.
Were there other players that didn't get playing time, but wanted to stay on the roster for 4 years? Yes. One guy in particular only had one pitching appearance in 4 years, and he got the out. He faced the league batting champion and got him to pop up. The bench went crazy. He just got promoted in an MLB front office analytics department.
Just my experience....