As nhmaf said, those white plastic intermediate gears will wear... especially when not meshing properly. The first gear, that drives the 1's digit wheel was a bit chewed up on mine, but not too bad... so I filed it smooth again. I'm pretty sure I've seen a set of those white gears for sale on the internet.
The drive sequence for the odometer is: metal gear, first intermediate gear, ones digit wheel, second intermediate gear, tens digit wheel, third intermediate gear, hundreds digit wheel, etc.
If the first intermediate gear on the odometer was chewed up but the metal drive gear was still stuck to the shaft, the trip meter should still work. That metal gear also drives the trip meter via a small brass spur gear. The small brass gear drives the large plastic wheel that is to the right hand side of the 1/10 white trip meter wheel.
There could also be a problem upstream of the odometer with one of the worm drive gears that get their rotation from the speedo cable... but I've never heard of that being a problem. The speedo magnetic rotor that moves the needle only has one set of gears between it and the speedo cable.
The only real way to tell is to pull the thing apart. You can gently chuck the brass fitting that the speedo cable engages with a drill and spin it COUNTER CLOCKWISE to see where the motion stops. While doing it by hand is possible, 1800 rpm at the cable equals about 60-65 mph equals 1 mile per minute. There is a whole lot of reduction going on in there!
I promised an article on my findings, and started to write one... just haven't gotten around to finishing it.