Hmm, possibly overheating? Open the case while it is running and make sure all of the fans that should be running are. If everything is fine, grab a can of compressed air and blow the dust out of them (while the comp is off this time). I would suggest downloading TAT or speedfan to check the temps, but your mobo may not have any sensors on it. My Dell only has a sensor on the HDD.
Could also be a PSU problem. Grab a unit from another PC and put it in that one and see if it changes.
Temps are fine. Its either psu, memory or HD. Download memtest, run it several passes. Hitachi has a nice HD test, dont have the link handy tho. If you have another rig around thatll fit your psu, try that as well.
Does it have an upgraded graphics card or any other power consuming upgrades? OEM are known for putting crap PSUs to run at the bare minimum. That or bad memory, if it doesn't have PSU problems, then run Memtest86 a few hours and see if you get any errors.
I just took a HP back. I had it for 2 days and it started doing this. I kept it two more days and read many posts on many forums that said the same thing. These included a lady who has had her's for a year, and it has spent more time being shipped back to fix it and them not fixing then it has with her actually working.
I took mine back. Shame too as I really liked the system the graphics were nice for what the kids needed. I decided to get out while I could.