In this case, they do. If you go to work for a competitor, the grantor of the options can simply revoke your options, claiming a violation of the non-compete. Then it would up to you to sue them and try to prove you're not actually violating it. You probably won't do that, so you'd have to decide if the hypothetical new job is worth the risk of losing the options.

I got a severance package a few years ago, with an associated non-compete. I absolutely did not violate the non-compete as long as the severance checks were rolling in, for the reasons I described above. Once the checks stopped, the non-compete became a lot less meaningful, because my former employer had nothing on me. They would have to have sued me, which probably wouldn't have been worth their time either.