As it turns out, my ATX 4-Pin Connector was fried.
Hopefully this information helps someone else troubleshoot their problem.
Description of reoccurring event:
Computer starts up fine (stays on without issue if left idle). Then, after 30 to 60 minutes of use, the monitor stops receiving a signal (and any audio that was playing stops); the computer's fans stay on and the hard drive keeps spinning (pushing or holding the computer's power/reset button has no effect.)
Hardware and Software:
Windows 7
Assembled desktop from individually bought parts.
Other details:
According to my processor temperature monitoring software, the computer isn't overheating.
I tried changing the hard drive (also running Windows 7), but experienced no change in the issue.
Does this problem have an obvious source? If not, what's the next step of troubleshooting I should take?

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