No. Or it is incredibly rare.
First, damages are routinely reduced dramatically on appeal.
Second, any attempt to recover beyond policy limits would be made against the hospital or other corporation.
Third, there is really no money in attempting to collect from a physician. It is the same as suing a patient for $200, there is no real money in it.
Fourth, it is not in the attorneys interest to try to collect.
A story from a book I read on Lord Mounbatten. Once he was visiting San Paulo, Brazil and they were scheduled to climb up to the "Christ the Redeemer" statue. He and his entourage were told to have 5 Reals to pay the robbers. Given that they were surrounded by heavily armed guards, they were surprised, but had the coins handy. Sure enough, half way up the mountain, a group of bandits came out of the surroundings, pointed guns at them, demanded 5 Reals each, but would not accept a penny(?) more. They faded away into the countryside and the group continued to the top of the mountain and came back down without any interference. The "robbers" were allowed to continue since they weren't doing a whole lot of harm. If they actually tried to really rob the tourists, or kidnapped them, the "robbers" would have been eliminated by the government - perhaps literally.
It is the same thing with malpractice attorneys. They know that if they push to hard it will lead to a physician revolt, with physicians banding together on expert testimony and draconian malpractice reform. There is an unspoken truce: "you don't come after us physicians personally, and we won't destroy your profession." As a result an attorney who attempted to collect above the limits would probably find himself beaten up by other lawyers as he walked to his car.
