From your brief diagnosis given it sounds like the Processor is getting too hot, and turning off.
(BIOS turns it off)
A Processor has a thermal limit. When it is exceeded the Processor turns off to keep from burning up.
(Temperature limit)
1) Has the inside of the computer, (To include the Power Supply) been cleaned out on a regular basis as needed?
Computer unplugged from power, computer case open, Anti-Static Precautions observed.
Use a can of compressed air for computers. Attach the plastic 'straw' to the nozzle of the can.
(Hold the plastic straw, and the can, when using to clean out the inside of the Power Supply. Otherwise the straw can come off of the nozzle, and go inside the Power Supply's case. {The straw is inserted into the ventilation holes of the Power Supply case)
Especially if we are discussing the 'Pizza-Box' desktop model here, compared to the Micro-Tower.
('Pizza-Box model case is open by pressing a button on each side, then pull the computer case 'hood' forward, and off)
2) If the computer has been cleaned out on a regular basis, let's go on.
2a) Check the action of the Processor fan. make sure it is constantly spinning, seems to spin at a rate that is deemed normal, (Not slow), and does not spin for a while, then stop, then spin.
[Typical Fan/Heatsink combo. May not look like yours, but is a typical illustration,
http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=3451342&CatId=493In the illustration shown the Fan is at the top, the Heatsink is at the bottom.
Typical construction of a Heatsink is a plate of metal that has tall, thin fins protruding from it.
The plate of metal absorbs heat from whatever object it is placed against.
(In this case the Processor. It is placed against the top of the Processor's case)
The tall, thin fins absorb heat from the plate, whereby they radiate the heat away.
A Fan is used in conjunction with the Heatsink, to use air to help carry heat away from the fins ]
2b) It may be that the Thermal Paste has dried up in-between the top of the Processor's case, and the bottom of the Heatsink.
Thermal paste, (Also known as thermal grease, and thermal compound. Not thermal Gel. That is a glue), is used to help transfer heat from the top of a Processor's case, to the bottom of the Heatsink.
The top of a Processor's case, and the bottom of a Heatsink are not perfectly smooth. A microscopic view would show 'Hills, Pit's, and Valleys'.
These imperfections create an air pocket.
Air is an Insulator, not a Conductor.
Thermal paste is a Conductor that fills these voids.
Should you wish to replace the Thermal Paste correctly, please state in a Comment.
To review:
1) Computer must be clean inside. To include the inside of the Power Supply. (A person can only clean the inside out so much, with a can of compressed air for computers. This however is better than it was)
2) Computer case open, computer plugged into power, observe the action of the Processor fan.
DO NOT put your hands inside. Only if the computer is unplugged from power.
[ I know this sounds like it's a reference to keep you from getting shocked. It is, but only partially.
You would have to put a metal object into the ventilation holes of the Power Supply, when plugged into power, to receive a bad enough shock. (Bad to Fatal)
It is more for the computer itself. You may fry out (Short Circuit) hardware components inside the computer.
The Power Supply receives either 115 volts, (US), or 220 volts, (Europe, Australia) from the main source.
(AC. Alternating Current)
This is converted into three main voltages for the computer. They are DC. Direct Current.
2a) 3.3 Volts
2b) 5 Volts
2c) 12 Volts
Two D cell flashlight batteries produce 3 Volts DC ]
3) Check the thermal paste. If a thermal PAD is used, take it off, throw it away. They are junk, and can't be reused anyway.
Thermal paste cost averages around $4 to $8. Depends where you get it.
The top of the Processor's case, and the bottom of the Heatsink MUST be cleaned Thoroughly.
Then thermal paste is applied Properly