I think this is what you have, if there are 3 wires connected to the switch. 

 

 

It seems like there is either a bad socket or a loose wire. Unless you can get the light fixture freed from the cabinet, removing the cabinet may be the only option.

Lew, thanks for the drawing!!!! The light that is out, has 3 wires going to it and their connection to the socket seems to be solid. I may be able to access the switch from the opening, for that socket, but I'm afraid of breaking the wiring, leaving me without any lights at all.

Your problem appears to be the bottom socket only

 Have you tried a lightbulb that you know for sure works, try the top lamp in the bottom socket.

With the switch off look inside the socket and be sure the brass contact tab is in place, try to pull it up or out to be sure contact is being made with the base of the lamp.

As others stated removing the cabinet is going to give you access to the sockets. Cut the paint around the cabinet before removing to keep the paint from peeling off with the cabinet.

The light fixture is bolted to the cabinet at the face frame.

Looks like I have the Rite/Left labels reversed on the drawing but the basic operation is correct.  @DuckSoup's comment about the socket connections would be my next thought, also. Sometimes the contacts in the sockets get bent over time and don't make solid connections. If you decide to go bend the socket contacts, you might want to turn off the breaker for the bathroom, for safety.

1 hour ago, DuckSoup said:

Your problem appears to be the bottom socket only

 Have you tried a lightbulb that you know for sure works, try the top lamp in the bottom socket.

With the switch off look inside the socket and be sure the brass contact tab is in place, try to pull it up or out to be sure contact is being made with the base of the lamp.

As others stated removing the cabinet is going to give you access to the sockets. Cut the paint around the cabinet before removing to keep the paint from peeling off with the cabinet.

The light fixture is bolted to the cabinet at the face frame.

Thanks, all of the above have been done except for removing the unit from the wall.

 It shouldn't be much work to remove the cabinet and solve the problem. I would suggest that before you pull it out have you wife pick a new paint color, damhik. 

FYI, medicine cabinets are trending in 2025. 

On 1/24/2025 at 12:24 PM, FrederickH said:

Thanks, all of the above have been done except for removing the unit from the wall.

And you HAVE checked the new bulb to make sure it works? VOE!

2 minutes ago, HARO50 said:

And you HAVE checked the new bulb to make sure it works? VOE!

Yes, I purchased 10 new bulbs of this type for any future replacements.

Now that all (or almost all) of our light bulbs are made overseas, I've had several that refused to light up right out of the box and at least a half-dozen that worked for a minute or two (or even less) before calling it quits!