I have a Dell Latitude D820 that I've owned for about 2.5 years. It is a Core 2 Duo T7200 2.0GHz, with 2 GB of RAM, an 80 GB hard drive and an NVidia Quadro 120M video card.
The computer was purchased in late November of 2006 with XP Pro, and included a free upgrade to Vista Business. (Vista was available on MSDN but not yet via retail, so the Vista Business upgrades weren't shipped until March of '07.) Since we had an MSDN subscription at the time, I installed Vista Ultimate on it pretty much as soon as I got it.
It ran happily until sometime in the spring of 2007 when Media Center (which I had never used except to watch DVDs) started throwing some kind of bizarre SQL (CE?) error. This error would pop up at random times just while using the computer. Furthermore, Media Center would no longer start. I never identified the cause of this error.
I had the Vista Business upgrade by this time, so I nuked the machine, installed XP and all the drivers, and then the Vista Business upgrade. Again, it ran happily for a few months and then started behaving badly once again. Vista Business doesn't have Media Center, so this exhibited completely unrelated symptoms. For no apparent reason and at fairly random times, the machine would suddenly appear to freeze up or run very slowly. For example, launching a new application window (any app) might take 30-45 seconds to paint fully. However, Task Manager showed very low CPU load, memory, etc. I tried all the normal stuff (chkdsk, defrag, etc.) and ran several diagnostic programs to try to identify any problems, but none found anything. It eventually reached the point that the computer was all but unusable, so I nuked it again and installed XP. This time I decided to stick with XP instead of going to Vista. However, within the past couple of months it has started to exhibit the same symptoms in XP that I used to see in Vista.
The computer is still under Dell warranty until December, but so far they aren't any help unless I can identify a specific problem.
A friend (partner in a now-dead business) has an identical machine that was purchased at the same time. His machine exhibits none of these symptoms, which leads me to believe it is a hardware issue, but I can't figure out how to identify it. Any ideas? Utilities? Seen something similar? At this point I can't even identify any pattern to the behavior, but would be willing to run a "stress test"-style app for as long as a couple days if I had any hope that it would find something.
EDIT July 17
I'm testing jerryjvl's answer regarding the video card, though I'm not sure it fully explains the symptoms yet. This morning I ran a video stress test. The test itself ran fine, but immediately afterward the PC started acting up again. I left ProcExp open and various system processes were consuming 50-60% of the CPU but with no apparent reason. For example, "services.exe" was eating about 40%, but the sum of its child processes wasn't higher than about 5%. I left it alone for several minutes to settle down, and then it was fine again. I used the "video card stability test" from firestone-group.com. Its output isn't very detailed, but it at least exercises the hardware pretty hard.
EDIT July 22
Thanks for your excellent suggestions. Here is an update on what I have tried so far.
Ran memtest86, SeaTools (Seagate), Hitachi drive fitness test, video card stability test (mentioned above). The video card test was the only one that seemed to produce any results, though it didn't occur during the actual test.
I defragged the drive (again...) with JkDefrag
I dropped the video card
