Desktop computer turning off, won't stay on very long either

Windows 7 64 bit. Installed win 7 professional fall 2010. Service pack 1.
Intel (R) Core (TM) 2 Quad CPU Q965 @3.00 GHZ

Had trouble with computer in March 2011 and eventually had to ship to HP with extended warranty (which ran out march 2012). They installed new motherboard and RAM. We upgraded fan (there are 2) and took RAM from 2GB to 8GM in May 2011. Has been running great, no problems til this weekend. It will shut off like being unplugged. I can't get it to stay on very long before it does it again. I am getting the sign on screen and the desktop shows up. I can do about 1 minute of operations before it goes out.

Opened up the tower and the fans are running, nothing feels hot.

I believe it has to do with hardware, so while I await other suggestions I am going to try replacing the RAM. Then checking the fans.

This is the boss' PC and when she ain't happy.... LOL

I have a desktop with the same problem before. And the funny thing was... the power cord to the tower wasnt plugged all the way in. Make sure its plugged all the way in. If it is then there a bug in your system that has a auto shut off kill command

started noticing the green light by the plug on the back of the cpu going off an on. Changed the cord for the power and we seem to be holding steady. Sometimes it's the simplest things! If tomorrow comes and we are out, I'll be back!

Well, I feel like it has to do with power.