I recently had a problem with my computer that drove me crazy for quite a while because it was intermittent at first. My computer was randomly shutting down/restarting. This would happen a few times then it would be ok. No viruses were found and the computer was fine when it didn't have this problem. Finally, it failed completely--no boot at all. The fans came on at maximum speed, but there was no display on the monitor. Powering off and cleaning the filters, which were already clean, had no effect, neither did dusting off the video card. I removed the video card and reseated it, then I powered the computer back up. It still didn't boot, but I now got a message on the monitor--I had forgotten to plug in the power cables to the video card. Powered down, and plugged in the power cables firmly. I then powered up the computer and everything worked! It continues to work! Lesson learned, make sure that when you are bundling your video card power cables that you do it in such a way that there is no stress that could gradually pull the cable loose. That was my problem, the way it was bundled was creating a pulling force which gradually loosened the connection, causing it to be intermittent at first, then finally, (over a period of weeks!) loosening enough to "fail" completely.
UPDATE: While it appeared that these steps fixed the problem, the problem came back! The computer was shutting down after about 45 minutes of on time, consistently. The power supply was replaced, under warranty, and that finally solved the problem! So the above were all symptoms of a failing power supply.