I'd like to bump this one because I'm a newbie and interested in the answer.

I would start by seeing if the alternator relay is firing, and cleaning the switches/observing contact on the relay.

Hopefully this one is not too easy, it happened to me when inspecting a machine for purchase.

Gottlieb Atlantis has been in storage for some time but has now been set up - it's a bit dirty and some of the switches need tweaking, but other than that it looks like it's in good shape. Except for one thing, you notice all of the 1,000 point opportunities award you points, but there's no chime. In fact, you observe no movement at all with regard to the chime. The chimes for 100 and 10 are working fine. How to troubleshoot?

This was the prevailing thought at the sale. The seller was a great guy, and an experienced electrician. He first determined that the contacts were all good, and then was able to determine in time that the coil was energizing. Still no movement or sound on the chime.

I didn't know much about the chimes at the time, but we're almost out of things it could be. What's left?

This is what it was! We flicked the plunger up and down with our hand and didn't notice anything in the beginning, but there was some grime on the plunger/sleeve that was apparently stronger than the coil. I am embarrassed when I think about how much time was spent on the electrical tests when all it needed was a cleaning.