I brought my dead home desktop computer to the office so I could fiddle with it after work while surrounded by large quantities of geek equipment that could step in to assist.
I tried to use a power supply from another computer in a sort of computerish version of jump-starting a car, but it was a hopeless endeavour because the cables on the power supply assume that the motherboard is right there. They barely reach the motherboard when the power supply is installed in the case; trying to connect one machine's power supply to another machine's motherboard is hopeless. I didn't want to unscrew the motherboard from the case if I didn't have to, so I left the power supply theory off to the side as a backup.
I yanked the memory from the machine—no help.
I yanked the video card from the machine—bingo. The BIOS made its happy booting sounds, but naturally it couldn't boot since it had no video card, no keyboard, no mouse, no memory, and no storage devices. But at least it went beep.
I went into my neighbor's office looking for a spare video card and found one on a side table. Plugged it in, hooked up the monitor cable, and... dead computer.
Oh, so close yet so far. It seemed that merely having a video card plugged in is enough to make the computer stop working.
I left my investigations at that for the evening. The next day, I told my neighbor about my "I thought I had it solved but I was wrong" close call the previous evening.
"Which video card did you use?"
— That one.
"Oh, I don't think that card works. Here, try this one."
Just my luck. I had run a video card test with a broken video card.
I returned to my office, plugged in the new video card, hooked up the monitor, and—Hallellujah—I got a POST screen and some angry BIOS messages since it didn't have a keyboard, mouse, memory, or storage devices.
Okay, plugged in the memory and storage devices, booted up the machine again, bingo, everything started up. Well, okay, I couldn't type in my password since I didn't connect a keyboard, but everything was working well enough that I was confident that I found the problem. Busted video card.
Thanks to all the commenters for your suggestions, even though I didn't act on most of them.
Bonus puzzle: When I ran the video card test that first evening, I unplugged a monitor from an existing computer and plugged it into the dead computer. After the test failed, I plugged the monitor cable back into the original computer. And the result was weird.
The picture on the monitor came back just fine, but it appeared to be frozen. I wiggled the mouse and banged on the keyboard, but the cursor didn't move on the screen and my typing didn't appear either. The computer was still running, however; I could ping it from another computer in my office, and I could even use Remote Desktop Connection to connect to the computer and use it. But the screen was frozen. It displayed a snapshot of my desktop at the point I originally yanked the monitor cable, sort of like Grandfather's Clock. The image finally disappeared when I powered the computer off.
When I turned the computer back on, the monitor showed just a blank screen. The computer itself ran fine, but now instead of showing a frozen image, it showed nothing at all.
The puzzle: Explain the exhibited behavior.