Also, using Cody's photo above, you see that dimple at 1-o'clock on the middle top ring? That's where the lamp socket makes contact. You can reflow that entire ring to get rid of the dimple and have a better mechanical connection. And sometimes I will pull the contacts on the lamp socket out a little more if the socket doesn't feel tight when it's twisted into position: more pressure = more scraping action = less oxide/gunk = better chance of a solid electrical connection.
I agree about reflowing all the connectors - that's something I'd do regardless. Also look for cracks in the PCB trace between the pads and the trace. And sometimes (this just happened on the backbox lights on my AFM) the wire will lift out of the IDC connector. It might not be visible, but it will lose electrical connection. Push it back in (ideally with an IDC tool but if you're skilled and/or lucky you may be able to do it with a screwdriver or needlenose pliers.
Long shot: I'm not sure anyone's mentioned it but aren't some (older design) LEDs polarity-sensitive? Try inserting the lamp assembly in both directions.