It could be the power button. You can use a paperclip to start it instead:http://www.broadbandreports.com/forum/remark,6763118
(You should be able to get the paperclip in the top. Practice with the power cord out first. If you can reach the terminals on the power switch and short them, that would be safer.)
If it starts consistenly, the power button is the likely suspect, otherwise it could be the motherboard, or the CPU fan not starting.