This may vary for cheetahs, but a good case study is european bison[1] who was restored after WWII from 12 animals, population now ~4.5k.

[1] https://en.m.wikipedia.org/wiki/European_bison

Last I heard the Iberian Lynx was down to about 70 wild cats, with about as many again in captive breeding programs. They were on the brink of caring about genetic diversity.

As I understand it, cheetahs are less genetically diverse than most big cats. (Sorry, can't recall where I originally heard about this; off-hand here is a non-technical reference: http://evolution.berkeley.edu/evolibrary/article/conservatio... .)