Bruce,
According to Mac Classic & SE Repair and Upgrade Secrets (ISBN 1-56609-022-9, page 193) by Larry Pina, a condition like the one that you described (also assuming that there is a startup sound) could be a bus issue related to a SCSI chip. Apparently, if in fact the problem, this could mean reflowing solder joints or replacing the SMD (surface mount device).
We should perhaps not rule out other things. It may not be a bad idea to have the logic board inspected. With the help of a magnifying glass, one can sometimes detect bad solder joints, leaking electrolytic capacitors and other faulty (for example, cracked or discoloured) components, etc. The 3.6 V logic board battery should be checked. Cables and memory modules can be disconnected/removed and then reconnected/reinstalled. A warning: All work inside a compact Mac is dangerous because of the high voltages involved, and must only be performed by someone with the necessary expertise. Charges may remain in certain sections even after the computer has been switched off and disconnected from the mains.
Jan