Case, motherboard and CPU? Did you try the usual leave the case cover off and point a fan at the open PC?
My brother's PC shuts down with no warning, and no pattern to speak of (shut down twice in 3 hours one night, then went 4 days without doing it). This PC was working fine for 6 months after being built, this just started happening out of nowhere. We've troubleshot just about everything we can think of to no avail:
Replaced the Hard drive
Replaced the power supply
Changed the surge protector
Plugged into a UPS
Checked the RAM (even used a new 4gb stick for a day, on its own. it still shut down)
Replaced the CPU cooler
Took it to a local PC repair shop (twice)
Stress-Tested the GPU (Even swapped the GPU with his old one)
Please help, we're not sure what else can be done and we're getting desperate. This PC has cost us so much money already, and we just want to get it working again.
PC Specs (if it helps):
AMD AM3+ FX 9370
AMD 990FX(AM3+) MoBo
EVGA GeForce GTX 960 4GB
G.SKILL Sniper Series 16GB (2 x 8GB)
Cooler Master Hyper 212 EVO - CPU Cooler
Corsair CX850M PSU

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic