Cannot Start Unless Powered OFF for 20 Minutes

Clarifications

It could be heat related. So open it up and have a good look and see if there is any dust in the case, if it is dusty then give it a hoover out but be very careful NOT TO touch any parts with the plastic end of the hoover.
The thermal paste on the processor is old or breaking off. Take off the heatsink and fan and give the processor a good clean and then replace the processor, then put on a very small amount of thermal paste onto the processor and spread evenly over it. Now replace the heatsink and fan.
PSU (power supply unit) is slowly giving up. Get another PSU and take out the old one then put in the new PSU. Make sure ALL of the cables are in their correct sockets and pin connectors.

Now start up the computer and see if there is any change.

Hope this is of help to you.

Want to test it? Unplug the machine, press the on/off switch and hold a second or two, do that maybe 5 times. That is, turn it on wait 2 sec, turn off again 5x. Now plug it back in and presto! it starts right up. There's a long complicated explanation why but just replace the PSU.

So try this to see. When you turn off unplug the system and press the Power Button for 10 seconds to discharge the Capacitors. If it starts after plugging it back in the most likely thing is that the Capacitors int he PS are on their way out.

While not 100% this is the easiest way to check but if it doesn't work by replacing the power supply the Capacitors on the M'Board are on their way out and you need to replace the M'Board.

For the best after market Power Supplies I always recommend Antec PS's as they offer far more protection tot he insides of the case from spikes and other rubbish on the Mains. But because of the superior protection they offer they tend to fail more often as they are designed to sacrifice themselves rather than do as the chiepies do pass everything into the case and degrade the electronics on the M'Board, HDD's and so on. They are also more expensive than the no name brands and considerably heaver. ;)

http://www.antec.com/Believe_it/product.php?Series=MjQ=

The Basiq will do at a pinch but the True Power are better.

If you want to know what size Power Supply that you require use the Antec Power Supply Calculator here

http://www.antec.outervision.com/

But remember that it was designed to use with Antec Power Supplies so don't use it's results for cheap no name PS's as they are not rated to that value for 100% of the time that they are being used. The cheap ones are rated to a Peek Value not a constant load so you will need a bigger one if you want to go that way. ;)

Col