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.
Actually sounds like a dying PSU. Often, power supplies will work intermittently when they are on their way out. Sometimes you can hit the power button multiple times and the PSU will work for a while, but yeah...try swapping in a new power supply, they're dirt cheap anyway.
Before a swap, I recommend testing it You can buy a power supply tester at a local PC shop for 10$ or so. This will show you if there is any problems with voltage on the PSU, but absolutely a good direction to check, especially if you know you have a low quality PSU.
My HP DV6000 has the ame problem of not powering up when I press the power button. It has not pulled the "turn on for 1 sec" trick yet but I defenitely has the " press power on button up to 20 times before it powers on" problem.
What sometimes seem to help is I take out the battery and disconnect the power, then it starts up. Does anyone have a permanent repair sugestion?