The at-fault party's bodily injury insurance is what would cover pain and suffering as well as lost wages normally unless you are in a no-fault state then your personal injury protection (PIP) coverage would typically be the first insurance policy to cover your medical expenses, lost wages, etc.

With regard to the limits of the at-fault party's bodily injury policy of 10/20, it is $10,000 per person for a total of $20,000 for all people injured in the accident. So you are limited to the $10,000 limit per person. The $20,000 limit would be if there were 2 or more people injured again each person would be limited to $10,000 each limit.

So for example if 2 people were injured each may get up to $10,000 for their bodily injuries. If 4 people were injured then it would still only be a total of $20,000 for all injuries so each person injured would have to split this amount. So if the 4 people were equally hurt the most each could get would be $5000 each, totaling all together the $20,000 limit.

If the person's insurance company pays the full amount covered by their policy limits, the insurer have no further financial obligation to you, the claimant. The at-fault party's insurance company will probably request that you settle for policy limits of $10,000 and sign a release saying no further claim will be made against the insurance company or policyholder.

Because an insurance company is not liable for anything more than the $10,000 limit named in the policy, they cannot pay any more to you, the claimant, for your injuries above this amount. If your injuries and property damages are way above the other person's limits you may want to seek advise from a lawyer to see if your state allows you to sue for monies above their insurance limits from the at fault person.