I remember feeling sorry for myself years ago when my thumb was in a brace -- a fellow student delivered a tsuki, and while trying for kotegaishi I took the punch with my thumb. Partially tore a ligament, permanently lost some range of motion.
While training with my brace on, two guest yudansha dropped by. One had a totally mangled or deformed hand, looked completely useless. The guy's aikido was kick ASS, in terms of his technique and in terms of his dive-right-in-there enthusiasm. I felt really inspired by his example and realized that (1) I have a lot to be thankful for, (2) while I am at all physically capable I will bring what I got to the mat.
Anyone who follows BJJ knows who Jean-Jacques Machado is. Possibly the very best BJJ martial artist on the planet. He has a deformed hand that can barely grab anything. Another source of real inspiration; that guy is a total legend.
So yeah, I believe it is possible to train effectively, where "effectively" means at your own personal edge. Where that is, that is personal to everybody and probably does not matter. The benefit is in pushing your own limits, consistently. IMHO anyway