This is where it REALLY helps to have a pc speaker (hooks up to the motherboard, not through the sound card/outside speaker ports) - I moved one from old system to old system for this reason, then I found some cheap piezo disk speakers for like a buck apiece to stash until needed. The speaker is used for the bios to give you a beep error code that can be really helpful. If you don't have anything you can get one from Radio Shack for $5 or so.
Recent 'dead' pc causes for me:
- power supply failed, swapped in a new one and it's been cranking for another 2 years. I was pissed becuase I try to buy pretty good power supplies and this one was only 4 yrs old.
- cpu heatsink pins are crappy plastic on socket 775 and have been breaking off over the years, I must have nudged the last remaining one loose and partially unseated the heatsink so it would hit thermal shutdown temperature in 10 sec to 30 minutes depending on room temperature. swapped heatsink and it's been cranking ok for months
Try to find out where the humming was - was it a fan that seized up? If it was the cpu fan or power supply fan that would shut it down quick - I've seen fans clogged with dust or pet hair, or just fail from age. SOME you can clean and oil, some it's just easier to replace...