I'd start simple and check the easy stuff like the power cable. After that, unplug any connected devices except the monitor (usb,hard drive...).
If no improvement, open the case, touch the case to ground yourself, then unplug the computer while keeping a hand on the metal of the case. Make sure all is secured and not loose, such as memory, video card. If no go, and if you feel comfortable, remove and reseat the memory. If no go...
Unplug all non essentials like video card, sound card DVD drive... and see if it will power up consistently. If so, put them back one at a time until it works if not...
Then it is most likely your motherboard. A smaller change is that it is your power button itself, but very slim. If you give me your model number I can show you how to bypass the power button and short the pins to turn it on.... but again, in most cases like this (and I have seen many ) the motherboard is at fault.
Let me know!