Unfortunately this is one of those rule out the bad component problems. You can first rule out the power switch on the case by trying to start the computer from the motherboard. If you feel comfortable doing this you can open the case and find the wire that connects from the case power button to the motherboard. It will most likely say pwr on it and have 2 wires going into one plug. The connection on the motherboard will have 2 pins. Once you find this you can take the wire off and take a screwdriver and "short" the connection on the motherboard by making contact with both pins at the same time. You don't need to hold it on there once it starts. Just hold it a second or two until it starts then you can take the screwdriver off. If it starts right away then you know it is your power switch on the case and you can fix it. I would just get a new case and transfer the parts if you know how or find someone that can maybe fix the power switch.
You can also try replacing the Power Supply. Its really easy to do. All it involves is unplugging the power connections off the motherboard, hard drive, floppy and CDROM. Then you take out 4 screws on the back of the computer and it will come right out.
Another option is to take out your cards such as modem, sound, network etc one at a time to see if one of them is causing the problem.
There is also the possibility that the motherboard may have a problem too. Unfortunately these kinds of hardware issues can be many things.