What do you do when you are in an accident that was not your fault but the other driver's insurance company won't pay for the damages saying they cannot determine who was at fault?

The insurance companies usually settle each claim independantly. Your insurance will pay yours, the other co will pay theirs. But your insurance will go up. If you have only the required liability insurance your company will not pay for your damage. If you do not have comprehensive and collision, you may have to sue the at-fault driver to force his insurance carrier pay. If you have coverage of your own, you can file a claim with your agent and immediately collect the damages (less your deductible) and your company will sue his if necessary. If and when they win or receive a settlement, you will get your deductible back. Also, if the other driver is determined to be At Fault, your rates should not increase.