Does it crash or spontaneously reboot after its running? If so then i would look at the psu like the others have said.
If not it could be as simple as a loose wire runnning from the power on lead of the motherboard to the front panel switch, which would explain it working when you furiously try to repeatedly turn it on.
 Open the case and trace the lead from the power switch back to the motherboard and make sure it isnt loose. Also check the connection on the power switch itself for any faults, loosness etc etc.
If you have a voltage multi meter you can check your rails by testing the supplied voltage on an empty 4 pin molex connector with the system powered on.
Its more than likely not anything else though such as ram, drives or addin cards. A problem with these unless it was directly and physicly shorting the voltage would not keep the system from powering on. It would simply keep it from making a successfull post screen and thus keep you from completely booting up.
Conclusion : Its the power supply or the power on/off switch wiring. Or possibly the circuit on the motherboard controlling the power on/off function.
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