First of all, in case this is relevant, my specs:
Dell XPS8300, Windows® 7 Home Premium, 64Bit, English
Processor: Intel® Core™ i7-2600 processor (8MB Cache, 3.4GHz)
RAM: 12GB Dual Channel DDR3 SDRAM at 1333MHz - 4 DIMMs
Hard drive: 500GB - 7200RPM, SATA 3.0Gb/s, 16MB Cache
The video card was an AMD Radeon HD 6870. As I'm going to explain, it has been replaced by a GeForce GTX 660.
I also use a APC Back-UPS ES 500 as an uninterrupted power supply plus surge protector.
So far I can tell, all of my drivers are up to date.
If any other spec questions are of interest, I'll provide that information if I can.
About two weeks ago, I ran into problems. At the time, I had the AMD card in the computer. The two specific things that were occurring were hard lockups (the computer was completely unresponsive, the screen was frozen, sound was making a BRRRRRRRRRRRRRR noise through external speakers that seemed to be the last instant of the sound it had been trying to play just before the freeze, and nothing from the internal speakers). The other thing was that on start-up, I was getting four beeps when I tried to turn the computer on, beep-beep-beep-beep. As far as my own research into Dell motherboard beep codes went, that seemed to indicate it was a memory problem. It should be noted, when I could get the computer to start up, it would be fine until I tried to do something demanding with it. If I could start it up, it could sit on the log-in screen all night without breaking a sweat.
I took the computer to a local repair place that came really highly recommended by my technically inclined friends (I suck and fail at handling repair of computer innards), and they were unable to replicate the problem, except that the screen was green tinted. An occasionally green tinted screen has been an ongoing problem for me on that computer, occurring whenever I updated the drivers for the video card; restarting the computer had always fixed the problems. Based on their advice, I replaced my video card with the GeForce GTX 660.
Taking the computer home on Thursday, I had a few lock-ups that night. No further problems occurred the next day or two, but on Sunday night I ran into issues again. I suffered from the hard lock-ups again, and then a Blue Screen of Death that blamed "Memory Management" (I can post a photo of the blue screen if requested). Then I had difficulty turning the computer on. The power button would flash orange and the computer would not start. I'd hear the fans rev up, but then they'd spin down and nothing would happen.
After one of the crashes, Windows ran an automatic test that found no issues. The repair place also reports that the memory and hardware is testing as working fine.
When the computer is running, I don't seem to have any fan problems. They sound fine (audible, but no worse than when I first bought the machine; I clean it regularly). I do not overlock, the temperature seems to be fine. I tried unplugging the computer completely, leaving it for a half hour, and then plugging in just the tower and the monitor to my UPS on the possibility that the other items plugged into my UPS (a second monitor, a laptop, and my speakers) might be causing power supply issues, but the computer still declined to start up, again giving me a flashing orange power button. While I had the computer, I did not think to check a different outlet. The UPS is currently working fine to power my 5.1 speakers, a laptop, and two monitors.
The repair place has the machine again, and they say they've retested it and turned it on 20+ times without issue. They are unable at this time to find any problem with the computer. Right now, my temptation is to ask them to replace the RAM (based on the memory management blue screen, the way that it seems to crash when I do something demanding, and the beep code I received earlier) and to replace the power supply.
(If I need to clarify anything, please don't hesitate to say so.)