Under traditional and justified common law principles, Gerard is correct: the availability of recovery from insurance or some other third source does not and should not affect recovery under the tort claim. These are two independent claims, in tort and contract. The insured has paid for the contract coverage (through private insurance, employment contract, or social insurance taxes) and thus he, rather than the tortfeasor, should obtain the benefit of that contract, just as he should if it were recovery under a Los Vegas bet regarding future injury or loss. This is the so-called "collateral source" rule. The insurer, however, may have (and usually does have) a subrogation claim against the tort recovery proceeds, by a term in the insurance contract (or the social insurance provisions).
However, this has been changed in many US states as one of the planks of so-called "tort reform," which allows the tortfeasor, rather than the insured, to obtain the benefit of the collateral source through reduced tort liability. Unjust and inefficient.