There are several factors that could cause the above problem in your diagnoses.
A) Processor Fan is not operating, or not spinning at the set speed
desired. (RPM. Revolutions Per Minute) Fan bearings are bad.
Replace processor fan.
Computer case open, observe the Processor fan as soon as you start the computer.
If the fan is not spinning, shut the computer down.
(Hold the Power On button in for a count of 10 seconds)
Observe if the fan spins slowly, or spins for a while, stops, then spins again.
B) Failing Electrolytic Capacitors on the motherboard.
C) Power Supply has a weak voltage power rail.
Failing Capacitors on the Motherboard.
Specifically the one's used in the motherboard Voltage Regulator Circuit.
A Processor must have a Steady, Clean, supply of voltage.
The tolerance range is Very small.
Too little, or too much voltage, the Processor turns off.
[BIOS turns it off]
Part of what the motherboard voltage regulator circuit does, is to regulate voltage for the Processor.
This link gives information about what Electrolytic Capacitors look like installed on a motherboard, and photos plus info, about the visual signs of Electrolytic Capacitor failure,http://www.capacitorlab.com/visible-failures/index.htm
(Computer unplugged from power. Observe Anti-Static Guidelines. Need
more info about the last sentence, please post in a Comment)
Requires either replacing the failing capacitors, or replace the
motherboard if bad capacitors are found.
If upon a close scrutiny with a flashlight, and magnifying glass, does not reveal visual signs of capacitor failure, then the next hardware component to check is the Power Supply.
3) Power Supply's have Electrolytic Capacitors also.
A computer power supply changes AC (Alternating Current) electricity from a household, or business, into DC electricity.
(Direct Current. A flashlight battery is an example of stored DC electricity)
Electrolytic Capacitors are used as Filters in a computer power supply.
They filter the incoming AC (Input Stage), and the outgoing DC (Output Stage)
Generally I have found that these are the weakest link, and are the first hardware component to fail in a PSU. (Power Supply Unit)http://en.wikipedia.org/wiki/Switched-mode_power_supply
There is a way to test the Power Supply.
Requires either a multimeter, or a Power Supply Tester, or using a KNOWN to be good, compatible power supply to substitute, and use as a test unit.
An economical multimeter can be purchased for around $10 to $15 at many stores. An auto parts store usually carries them.
An economical power supply tester can be purchased for around $20.
This is an example of a power supply tester,http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=5250576&CatId=5471
Computer dirty inside, as well as the inside of the Power Supply, will cause detrimental factors to the computer.
1) Dust is a Static magnet. Dust can create pathways for static electricity to follow, and the DC electricity used for a computer.
This can cause hardware components inside the computer to Short Circuit. (Fry out)
2) Dust, dirt, hair, etc, can clog the cooling components of the computer.
The computer case fan/s, the Processor fan, and Heatsink, and the Power Supply fan, and Heatsink/s located inside the Power Supply.
Air is drawn in through the front of a computer, (Or on some custom PC's through the side), and flows over the hardware components inside the computer, helping to cool them.
If a layer of dust, dirt, hair, and so on, has coated these hardware components, the heat will be trapped, and the air flow will not be able to carry the heat away.
If the computer case fan's are coated, their cooling capacity drops tremendously also. The fan, or fans, will not be able to provide the air flow needed.
If the Processor fan, and Heatsink are coated, the cooling capacity of these two components drops tremendously.
A Processor has a thermal temperature it can operate in, and will turn off if the temperature is exceeded.
It's a Fail Safe feature that is built-in, to keep the Processor from burning up.
[A Heatsink is essentially a plate of metal that has tall, thin fins protruding from it. Heat is absorbed into the plate, whereby it is then absorbed by the tall, thin fins.
Air flows in-between the fins, and helps carry the heat away.
This is an example of an average Fan/Heatsink combo,http://www.tigerdirect.com/applications/SearchTools/item-details.asp?EdpNo=2604150&CatId=493
The Heatsink is the aluminum finned part sitting under the fan.
Using a can of compressed air for computers, on a regular basis as needed, can help prevent many computer failures.
1) Check the Processor fan. If the computer is dirty inside, there is the obvious factor for it's failing.
2) Check the capacitors on the motherboard visually.
3) Test the power supply, or substitute it with a Known to be good, compatible unit for a test.
Must have at least the same amount of Wattage, or more.
Must have all the correct power cables needed.
Must have enough of the correct cables]