Yes, if you hit analog tape with enough level, then the tape saturates and you get what is commonly referred to as "tape compression". It's really not that dissimilar from amp compression in some ways - the medium can't handle any further increase in level, so the waveform peaks start getting rounded off...
The funny thing is, nearly every time I'm in LA in one of the big studio complexes, all the control rooms that are booked have Pro Tools systems in use, and the hallways outside are lined with unused Ampex and Studer tape decks...
It's harder to use those pitch correction plugins on tape.