Is it my Power Supply or Heatsink that is failing?

Could you post more system specs? It's very hard to work on the information you provided  
what I can say is that it's probably your PSU (power supply unit).

Brand and model of PSU?

Load a Linux LiveCD. See if it's a Windows problem.

Test the memory with memtest86+.
Run manufacturer hard disk diagnostics.

First off, I want to thank all of you for the quick replies, and also apologize for including such little information.

System Information:

OS: Windows 7 64-bit
CPU: AMD Phenom II X4 940 (Current temp 35C)
RAM: 8.0gb OCZ Dual-Channel DDR2 @ 401MHz 6-6-6-18
Motherboard: ASUS M3A76-CM AMD 780G Chipset
Video Card: GTX 260 (current temp 50C)
Hard Drive: Western Digital 500gb

Nothing is over clocked, and I would again like to stress that this problem ONLY happens while playing video games, or video play back and that it is sporadic when it happens. I am 95% certain this is not an OS problem.

I can think of two possibilities:
A) Driver problem. Update to the latest Nvidia driver. But, before you do, remove all traces of any video drivers.
Here's how:

1) Download driversweeper to your desktop, or wherever you can find it easily.
http://downloads.guru3d.com/Guru3D---Driver-Sweeper-(Setup)-download-1655.html

2) Download your new driver to the same place.
3) Add/remove programs, uninstall any Nvidia or ATI vga drivers and programs.
4) Boot into safe mode. Press F8 repeatedly while booting to enter safe mode. Your display resolution may be minimal.
5) Run driversweeper. Check the selection boxes for the removal of all nvidia vga drivers and ati drivers.
6) Rerun driversweeper. All such components should be gone.
7) Install the new driver from the desktop or saved location.

B) Power problem.

If your 600w psu is old, or of dubious quality it could be at fault. A video game pushed the GTX260 into 3d mode which wil require more power than normal. If the psu can not do this reliably, then you will get artifacts and other strange symptoms.

See if your PSU appears on this llist of psu quality:
http://www.eggxpert.com/forums/thread/323050.aspx

If A) above does not resolve the issue, see if you can borrow a different psu to try.

Idle temps are fine, so I would assume load temps are normal too. But to be sure monitor your temperatures during a stress test (CPU and video).

@Geofelt: I have already done similar, and I am using the latest drivers.
@Scudst0rm: I have seen the CPU sitting at 50c right after a crash.

Are you talking about the cpu heat sink or the GTX260 heat sink?

It would seem that the cpu is cool enough. 50c. is OK.

If the GTX260 heat sink is faulty, the card will heat up under stress and cause artifacts. Is the GTX260 fan operating?

I case cooling possibly an issue? try running with the side covers off and see if it helps. Run a housefan directed at the vga card to give it plenty of cold air.

I have been talking about the CPU Heatsink (running at 50C under load) My video card is sitting at 50C right now, although the last time I checked the fan was still running (this was not before the problems started) As for the case it sits around 25c inside, I have 2 massive fans on the side keeping things cool.

Oh, I should also add that while the freezing is new, the video card not turning the monitor on was an old problem, and predated the freezing. For a while I simply switched to on-board graphics which worked without problem (except for being terrible) because my video card would simply not turn my monitor on.

It sounds to me like the video card is faulty. If it is from EVGA or XFX, they have a lifetime warranty and will replace it. Check that out unless you can test it in a different PC.

A low quality PSU can deteriorate over time; that is a second possibility.

I'm guessing a deflective power supply but it's like a 50/50 chance unless you ran furmark and it is stable, then it's 100% on psu

What brand PSU is it? There is a lot of variation between PSU brands at a given wattage. If you can pass prime 95 for an hour, and you can pass furmark for an hour, and the combination of the two makes you crash then it is almost certainly a PSU issue. Its a fairly simple and free way to check.

I want to apologize for taking so long to respond, and I hope I have not lost the people who have been awesome enough to help.

I ran fumark at 1900 x1080 (native resolution) and my computer lasted a grand total of 88 seconds before freezing. Videocard reached 78C before it stopped reading. I have not run prime95 yet, but I am about to.

EDIT: I have no idea how to use Prime 95  

Prime95 is easy. Just start it, it will load up all threads at 100%
Select as an option rounding checking. Let it run long enough until the cpu temperature reaches it's maximum and stops increasing.
Start up cpu-z and monitor the cpu multiplier to be certain that the cpu is not throttling itself.
You should see NO errors on the prime95 report.

You might also want to test the ram with memest86+ for a full pass. Again, you should see NO errors.

Just wondering, what is your opinion on the fumark results?

Also, while I understand the concepts behind your words concerning what prime95 is supposed to do, I am however not aware of how to get prime95 to do this for me. So far I opened it up, and fiddled around with the torture test option.

Thank you for the prompt response.

I am not familiar with furmark. I suspect the vga card is defective.

On prime95:

Where it says run a torture test select which test. the default blend should be OK.
clock OK and you should see a list of messages from each of 8 worker threads.
Under advanced, clock on round off checking.
If you open task manager, you should see all threads at 100%.
If you started cpu-z verify that the cpu multiplier stays at it's maximum. If it dips, it is because the cpu is throttling to protect itself.

Just monitor the messages for any errors. If your cpu is unstable form OC or any other reason, you will see threads stopped, or rounding errors.

Keep it up until you are satisfied it is stable.

To stop the test, the test option has stop and exit options. stop temporatily stops execution, exit closes the program. You want exit.

I use the speedfan temp monitor, and I can select a cpu temp chart which shows how hith the cores are, and when they reach their max. It might only be 10 minutes or so.

I ran the test and kept an eye on the cpu temp and multiplier for 10 mins.

Prime95 mentioned no errors, or anything to alert me of any problems (I stopped around test 7)
Core Temp and CPU-Z both stopped at about 57C
The multiplier never throttled down

The cpu is fine. Cooling of the cpu is good.

Looks like it is the vga card to me.
If you can borrow a psu to test, that might be a good idea to exclude that possibility.

Just out of curiosity, have they released any cards that are better than the gtx 260 but with a similar price point? I know little of the new ATI models, is the 5770 or the 5850 model comparable?

the 5770 performs similar to a 4870 which perform similarly to a GTX260, they are around the same price point, but with supply stabilizing the 5770 should drop a bit soon.

Annnnnnnd i'm back.


Changes to computer:

New Video Card, ATI 5770
Now using on-board sound after removing sound card.

Current symptoms:

After few minutes in a game (tf2 and bioshock 2 are what I have tried) computer screen will go blank (except this time there are feint black bars running vertically across the screen, I do not think this means anything more than the solid black but back-lit screen I had with the nvidea card)

Sound still continues playing in the background (no dynamic sounds just background sounds, like the bioshock 2 music will keep playing or Pandora will keep going, have not tested to see if it will load another song)

Keyboard commands make no noticeable change to the sound, can not hear the character move or act, pressing escape does not appear to bring up menu (based solely on expected sound changes) This is the same as before

Single press of the power button will not shutdown computer, can only power down by holding down the button

Tests

FURMARK: 1900 x1080 full screen | MSAA 8x | Video card maxed at 70-71c, there was no crash this time (old card did not last 30 seconds) CPU ran at 36-38c

PRIME95: No crash, same as with old card, cpu did not break 50c, multiplier did not throttle down.

1. jcs and hunter both asked what brand and model power supply you have. I do not see a replay. Please post the information.

2. The feint gray horiontal lines you are seeing with the 5770 video card is a problem that has been well documented. Please go to either the ATI web site or the video card manufacturer's web site. Download and install the patch/fix.