New York is a comparative negligence state. This means that liability is based upon percentage of fault. For example, if you are a passenger in a car that has an accident with another vehicle, and it is determined that each car is 50% at fault, then each respective insurance company will generally pay for 50% of the value of your case.
New York is also a joint and several liability state. This means that if one party is at least 1% at fault, you can get the full amount of your damages from that company and it will be up to them to try to get paid back from any other liable parties. This happens quite often, when one car has a majority of fault but only carries minimum insurance limits and another car has less fault but carriers much more insurance.