When this happens to me, it's often a peripheral/drive malfunction. Unplug EVERYTHING, including floppy/CDROM/hard drives, both internal and external, and plug in only the keyboard and display. That helps to narrow it down a bit for me, and usually works.

Otherwise, start eliminating everything else that can fail: expansion RAM, accelerator devices, expansion cards, etc. Get your machine completely basic.

If none of that helps, look for damaged components on the logic boards, particularly ICs that look a little toasted or leaking capacitors.