It doesn't appear to matter how many radiators I put onto my base, or whether or not Kerbal Engineer tells me that I'm in the clear as far as waste heat goes in the VAB. My nuclear reactor on my base always overheats, even when my resources menu shows that I have zero waste heat, and eventually my radiators explode.

Am I doing something wrong, or is this a bug?

Yeah, that is acutally the problem. The heating mechanism of KSP is quite buggy. There is not much i can do about it. I would suggest using NFE when Nertea found a workaround for his reactors. The reactore of KPBS uses the modules from NFE if it is installed.

This rather sounds like a bug to me. But i did not experience it yet myself or heard from another person from it. Can you post some details? E.g. a screenshot of the base the craft file for the base?

Is it is happening when you are coming out of time warp, I believe there's a bug in stock KSP that causes all the accumulated Heat there is not calculated during time warp to be applied to the reactors at the moment you come out of time warp.

I don't know if that is still a bug, but Nertea was talking about it a few months back in the NFE and Heat Control threads.