I have a 2002 Montero XLS with 130K miles on - only the lower right
light on the dash lights works no matter where the transfer case lever is
2H, 4H or 4L. I suspect that the lower left light bulb is burned out but
I believe that the car is not shifting into 4x4 no matter which position the
transfer case lever is in. Any suggestions as to what might be the problem?

 

I had the same issue with my 2003 LTD. It was simply the dashboard bulb. Dealer took care of it under warranty.

 

first, try just changing the bulb...
If that doesnt work...
put it in locked 4wheel drive mode, and jack it up.
place jack-stands under it for safety, and so that you can get all four wheels off the ground. have someone climb in, with the car still running, and still in 4 wheel mode... tell them to GENTLY apply a SMALL amount of throttle until the tires start spinning. If you can only see the rear tires moving, then you have your answer. If however you see both front and rear turning, then you have your answer. Also, Just because they are all turning does not mean that they are functioning properly. Your center diff. could be out, and sending enough power to spin all four wheels in the air, but only routing power to the least grippy wheel when you need it.
does that make any sense at all?