[fuusnzvv] Prime enough

You come across a composite number that resists factorization, so you call it prime and use it accordingly.  Under what conditions or applications is this a bad idea?  Under what conditions is it fine (so long as no one discovers a factor)?

Probably a bad idea if you don't know the factorization but someone else (secretly) does, though I have no concrete examples of how things go wrong.  Any other cases?

Previously, if you can't factor N, then factor N-1.