It looks like in the uvmap you have "folded over" some mesh regions onto the same skinmap region (e.g. the fingers in this model). I can only guess that the "meshing" algorithm used by GLQuake and its derivatives doesn't take this possibility into account somehow. (DarkPlaces doesn't use the old 1997 algorithm.) I don't know anything about this meshing though, so I'm not sure why it doesn't work.
I'm pretty sure that you'll have to either re-unwrap and skin the model without using these 'shortcuts', or the engine's meshing code will have to be rewritten. I would recommend re-unwrapping anyway because the skin doesn't need to be this detailed on the PSP, and there's a ton of wasted space anyhow.
Overlapping two polygons on the same skinmap region should be fine, but I'm not sure if any of these overlapping polygons might actually share an edge, which may or may not be bad. I don't know.
Perhaps I'm not even on the right track. The fingers weren't the only part that were screwed up, the rest of the hands were too, and I don't see why that would happen.
(BTW reducing the skin width/height doesn't fix it.)