XPS 200 - Sudden Shutdown Problem - Please Help

I am having problems with an XPS-200 (Pentium D) shutting down without warning or explanation. If anyone feels they have information that may help me to diagnose and/or resolve this issue, I would be very grateful for your time.

The system worked fine for over two years and then this problem began about a month ago. So far it does not seem to be progressive, the shut downs happen about as frequently now as they did when I first noticed the problem. They do seem to occur most often during periods of higher-than-average computer activity, but I can't say whether it's linked to processor activity, disc activity, memory read/write activity or any other specific system component. If I start editing videos in Liquid Edition, the system almost always shuts down with a few minutes of use. If I'm just browsing the web, it may not happen even once in six hours.

Please note that I am using the term "shut down" very loosely. The system is not shutting down or even powering off. When it happens the monitor suddenly goes into power save mode and pressing the CAPS LOCK button (corresponding light does not come on) reveals the computer has failed, as opposed to just losing the video signal. All disc activity stops and, interestingly, the green light on the DVD drive comes on as solid green. This has increased my worry that perhaps there is a problem with the on-board SATA controller, though it's pure speculation. The power light on the computer remains solid green like when the computer is running normally and I'm pretty sure the system's fan keeps going at whatever speed it was when the failure took place. So it's like the whole motherboard seems to die. Whatever is failing, it has to be something so basic that everything else just stops dead without it. Could it be a problem with the CPU itself?

Here is one more interesting point that hopefully won't prove to be too embarrassing. If the problem starts happening constantly, as it occasionally does, picking up the computer and shaking it around definitely seems to have a positive effect. I know it sounds crazy, but it's true. It will shut off six times in a row before Windows even can load. Then I'll shake the computer and it will boot normally, with no return of the problem for as long as 90 minutes. (When I use the word "shake", rest assured I am speaking of an activity that is not extreme enough to cause any damage.)

Can anyone recommend the next steps to take in establishing what the problem is?

I have reseated basically everything on the mother board including the CPU itself. I have tried entirely different memory modules (borrowed from another machine). I have thoroughly cleaned the inside of the case, the CPU fan, the power supply fan and anything else that seemed dusty. None of these actions has helped or worsened the problem in any predictable way.

Please communicate your ideas! Thank you!