Can you get a tax exemption for money paid for a deductible on an accident claim if the accident is not your fault?

Prefaced by the standard - generally, under most circumstances, etc - and what your looking for is a deduction (of the expense), not an exemption (which would be advantageous if it was income). Yes, as it would be a casualty loss deduction. However, as the casualty deduction has some restrictions to meet before it can be claimed, which is likely not to be met by the generally modest amount of a deductible, unless you have losses on top of the deductible, it is likely not really available.