I tried a case in JP court of all places late last year. It was a subrogation case where the insured was hit by another, my client (insurer) paid for the damages and went after the other driver. It seemed somewhat straight forward, but my involvement started just before trial. There was little to no discovery.
When I got to court, I visited with the other party, who was unrepresented. Turns out, he was indeed covered for the accident by State Farm. Most of these cases involved uninsured drivers -- and in many cases, my clients have terrible trouble serving them because of who knows where they are or go after the accident. Anyway, this guy refused to turn the claim over to his insurer because, in his mind, he did nothing wrong and didn't want his rates to go up. I told him he HAD to turn the case over to the insured - there are ARGUABLE legal theories supporting this, even if they can't stand up to scrutiny at a higher court - and I made the same argument to the judge. However, the judge wouldn't make him do so and wouldn't continue the case without both of us agreeing and the defendant would not agree.
Result? As expected, I easily got him to admit that his was the vehicle that hit my client's insured and in spite of his belief that he did nothing wrong, the car he hit was in a place she was legally entitled to be. I obtained a judgment against him for several thousand dollars. I had told him prior to trial that if that happened, State Farm would likely deny his claim if he then presented it, because by contract, they had to be involved in the entire litigation process.
Points: 1) you can't go on your BELIEF of what the facts or the law are; and 2) he was a fool because all he could see was a potential raise in insurance rates vs. the many thousands he would be on the hook for if he lost. He didn't foresee a loss because, in his mind, he did nothing wrong. (See number 1).
Other than criminal lawyers who try cases almost monthly (or did, up until recently), lawyers that handle insurance issues are some of the best trial lawyers there are -- for the same reason: they've got a LOT of trial experience. Let your insurance company handle anything they're obligated to handle and worry about rate increases later.