Sometimes, when I'm playing a game, it will just give me a black screen and the sound will cycle erroneously. Then, while still sounding, I'll get my monitor's default "off" screen, as if my PC has turned off. Does this seem like a fail-safe from overheating?

If you don't have speedfan installed then now would be a good time, so you can monitor temperatures. Enable the graphs in it so that you can come out of a game and check the temps =o

If Speedfan can't read your GPU temps, get GPU-Z. Also has a graph (sorta) feature that will let you see changes over time =o

I forgot what parts you have, but a rule of thumb is 60C as a limit for CPUs, and 90C as a limit for GPUs.

It's not a heat issue. My temps never go above 55C on either of my CPU cores, so I'm ruling that out for now. I think it might have something to do with the fact that I'm running too many things at once. The games that seem to do it are strange, though. Borderlands and Dragon Age give me no problems, but Trine and Venetica do. Odd.