I've actually had a very similar problem a few times before, and each time it was due to
capacitor plague.
It's fairly easy to diagnose: you just look at the capacitors on the affected piece of hardware (the cylindrical thingies that stick out) and check for signs that they've been under high internal pressure.
This can be in the form of bulging tops of the metal caps, the entire metal cap being pushed off slightly, or complete breaches that result in brownish goo leaking out. More details can be found in the wiki article above, and I recommend reading it.
Since the capacitors regulate the power in the computer, a failure in one can cause increased load to be put on other capacitors, leading to more failures. If it's on your motherboard, this means that other components (e.g. video card)
could have developed the same condition, even if they didn't have defective capacitors. In other words, check all your cards and boards for this condition if you have it.
If you've got it, you have two options:
1. Replace the affected piece(s) of hardware - This is the only option for most people.
2. If you know an electrical engineer or someone who has a lot of experience soldering and repairing electronics, get them to unsolder the capacitors and replace them with equivalently rated new ones.
There is no guarantee that if you do this that it will fix the problem, and it will usually be pretty rare to find someone who knows exactly how to do this. If it works though, it can be worth it to save some money on new hardware.
In the last 5 years, I've had this twice. Once on a motherboard, which caused it to be unstartable, and once on a video card. It was only worth it to fix the video card, since new mobo/cpu combos at Fry's go on sale all the time.
Even if you've decided to replace it, check your motherboard as well as your other components. It's more common than you might think.