Sometimes a quick visual inspection of the interior of the computer can lead to the cause of the problem. Double-check the cabling, cards, memory, etc. to make sure that everything is secured in place. Even if the cards appear to be fine, I've seen it where they sometimes need to be removed and reseated. Don't forget about cooling as well. Make sure that the system has adequate cooling, that the existing fans/heatsinks are not clogged with dust and have good mobility with the flick of a finger. Double-check the fans are operational with case open and system is powered, and most motherboards have basic temperature monitoring for the CPUs and speed monitoring for the fans. On the motherboard, make sure to check the capacitors. Over the years (as recently as a couple weeks ago), I've had to replace motherboards because the capacitors had gone bad:
see -- http://en.wikipedia.org/wiki/Capacitor_plague
Some people have already mentioned it, but it needs to be stressed, a *good* power supply is mandatory and if necessary a UPS. The power supply can be perfectly operational and even pass with a power supply tester (also a good investment), but if the power being supplied to it is not consistent (brown-outs) or simply not adequate to drive all the components (e.g. video cards, # of drives, etc.) that can cause problems. In one case by simply swapping the cheap power supply out for a good quality one that I had as a spare from an older system resolved the problem.
Inproper BIOS settings can also cause problems. Memory/CPU voltages or speed may be incorrect? Conflicting on-board video/audio still enabled when add-in video cards and audio cards have been added?
I still haven't even gotten to the software debugging side of things...
* UNIX is a Trademark of Bell Laboratories.
