What happens when the insurance limits aren't enough to pay my bodily injury claim?

I was a penger in an auto accident in June where my arm was broken. What happens if the at fault person's insurance and the underinsured limits from the car I was in is still not enough to pay my claim? Does it then become a personal lawsuit? I think it will come to that because my doc says this injury is one that may not heal. I'm praying it does, but if it doesn't he's going to have to do surgery again. That wouldn't be until next year and I would have to go through all that pain of healing all over again.