Hmmn, yes that is a strange one. As Sandra said, I have often seen some faulty wires from the power button, as well as the wires not being plugged fully on the mobo.
The button itself may be causing the issue if its cheap and nasty.
Locate the POWER SWITCH connection on the mobo - should be two pins. Remove the wire that goes to the power button.
Use a screwdriver to "short" these pins - effectively doing what the power button does. Its safe enough, just make sure the screwdriver doesn't touch anything else!! Should be easy to do.
If the mobo powers up everytime successfully with this method, then its mos def the wires or button.
If not, then there is another issue! It COULD be related to capacitors on the mainboard, which would be bad news really coz you have to replace....
good luck