However, it was my understanding that not all natural diseases grant immunity to survivors. I'm not an expert, of course.

I've been led to understand that this was usually the other way around, or that the mechanism that allowed their survival in the first place was "change something in the immune system, see if it works, repeat until it does". Through some magical process of biology or chemistry afterwards, the found solution is then "remembered" and ready to be deployed again if the disease returns. I'm not quite sure whether anyone understands the exact mechanism behind this magic, but I certainly don't (yet). *

By "the other way around", I mean a selection effect; they survived because they were already more resistant and had the right biological configuration ready to become immune to it or somesuch. I'm not clear on the details, this is all second-hand (but from people who knew what they were talking about, or so it seemed at the time).

* ETA: Got curious. Looks like there's a pretty good understanding of the matter in the field after all. +1 esteem for immunology and +0.2 for scientific medicine in general. And those are some really great wikipedia articles.