When I push the power button the computer seems to be starting normaly but then shuts off after 1 or 2 seconds. Nothing on the screen during this time. It will still succesfully turn on about 1 in 10 times and then have no apparent problems, although the success rate seems to be dropping.

My computer knowledge isn't great so any help would be appreciated.

This is a heatsink issue:

The metal of the heatsink is moving very slightly... most of the time shorting the board, putting power where it shouldn't be, and then on occaison the power goes somewhere useful and it boots. Either change the heatsink, or remount it.

I used to play the bootup game too. And I had to go through 3 different heatsinks for a certain board and then sequentially trashed it.

This is also a rare power/motherboard issue, but its nature is very aimed toward something metal hitting the board on the backside or front. I also recommend removing the motherboard and moving it onto carpet/foam to test shorting from anything but the heatsink.

Best of luck, dood.