I recently found that my computer would not start, when I press the button on the front panel. The solution…
The workaround was to turn off the computer at the back, using the switch where the power cable comes into the computer. I guess you could switch off at the wall too. And then after waiting 30 seconds, I found that the computer would then turn on.
This post “Computer won’t start unless I flip power supply off and on again…” suggests that this could indicate that either the power supply unit (PSU), or motherboard, could be developing a fault.
My computer is 3 years old, and I guess a new PSU may be the cheapest option to try out.
Feb 2012 update
The computer failed to start-up at all, and the workaround did not work either. The solution, as expected, was to get the power supply unit (PSU) replaced. It took a morning and cost GBP30 ($50) including parts and labour, but now everything works fine.