It's timed entry; Access the Cloisters at a specific time. Do you have to leave the Cloisters at a specific time?

No, other than closing time of course... :) It's a relaxed and peaceful atmosphere.

Timed entry just ensures that they don't go over capacity for certain blocks of the day but you can stay as long as you'd like

I didn’t reserve a time nor had to leave at a certain time. Once you get the tickets you go in and enjoy for as long as you want. I went on a Saturday afternoon and there were plenty of people and I was never told I had a certain time to leave.