I think I may have discovered the problem, but I want to know for sure, so I'm asking to see if anyone knows.
Some of the mesh parts were uvw mapped as one piece but afterwards I applied an Edit Mesh modifier to detach faces as a new piece.
The pieces missing caps were at the places that I had detached the faces. (I unwrapped the legs with the hips and the hands with the arms. The caps show up for the legs and hands but not on the arms and hips where the faces were detached.)
If this is the problem, where should I move the Edit Mesh modifiers in the stack to fix this? (I'll experiment to try and figure it out myself, but would appreciate feedback from anyone that knows for sure.)
If that was it, that was a really deceptive problem.
Luckily I find them all.