Yes, you could get that sort of reaction if your CPU heatsink and fan weren't working properly. Most modern MoBos will protect the processor at the very least.

I had a problem with one of my other machines recently. It would give me random shut downs, but mostly quite soon after it was booted. Turned out to be the power supply fan wouldn't start properly sometimes.

Be sure to clean off the old thermal paste and use the silver stuff (antec, artic or whatever) .......... like I said 50c is on the high side unless you are overclocking