Well, I reseated the graphics card but that didn't seem to do much. I did discover, though, that there was quite a bit of dust on the graphics card in places I could not reach easily when I was dusting everything yesterday. See:
http://img440.imageshack.us/img440/4...7199038333.jpg . I didn't think to take the graphics card out yesterday, but now I know better -- I'll have to buy another can of air soon and clean the bits where I missed.
The good news is that I did end up fixing the problem -- my machine's been on for about 35 minutes so far (typing from it atm), whereas before I could barely two minutes use out of it. Unfortunately I don't know what the problem was exactly, as I replaced the cable leading to the monitor AND I used a different auxiliary power connector that happened to be unused inside my case, and only then did I test to see if it worked. I have my doubts it was the DVI cable's fault, but I'm in no rush to start pulling my computer apart again to try and isolate the problem cable/connector though. In any case, thanks for all your help!