Welcome to Tom's Hardware,
Basically what I think you have going on here is thermal shutdown.
I would first take all panels off the unit, including the front panel and CPU vent and blow everything off really good with compressed air. Very important to make sure you get all of the dust out of the heatsink fins. A soft paint brush can help.
The other thing I will point out here is, you do have an older computer and after about 7-10 years things start drying out. After ensuring everything is dust free, you can replace the thermal compound on your CPU, here is a video that explains how (ignore the replacing CPU part and that he is using an aftermarket heatsink - the principles are the same) http://www.youtube.com/watch?v=I3gx6c62D7I
if you have any questions or concerns, ask first for clarification.
Last and worst, since your computer is old (near if not over life expectancy) it may just be dying (micro fissures in the mobo that heat up and lose connection, capacitors that fail, etc...) but give it a good cleaning first, see if that helps.