It's an odd problem as I would have half expected some kind of intermittant fault whilst the machine was running if it was indeed a short somewhere along the line.

Having said that, it should still be possible for you to test some of the hardware in your case - You're not guaranteed to find the culprit but it at least means you won't need to buy any extra hardware and/or equipment.

If you've got more than one stick or RAM, try taking one out and seeing if that works, If not, swap it and try the other one in bank one. Also, disconnect any hard drives or DVD/CD-ROM drives one by one before rebooting and see if that cures anything. If you have any non-critical expansion cards like sound or network, remove those one by one as well.

Like I said, you may not find the problem if it is graphics, CPU or motherboard related, though you can at least rule out the above hardware.