Should Costco pay for repair to my car?

I bought a set of tires from a warehouse dept store (Costco) a while back. With the tires, they offer free rotation for the life of the tires. I brought the car into Costco yesterday for rotation, and the technician broke off one of the lug nuts because it was frozen. He called before he did it and warned me that it might break if he tried to remove it, and I gave him permission to remove it. When I examined the broken lug, I noticed that he was able to unscrew the nut about 3 threads, and that it appeared to be cross threaded. I think Costco should pay for replacement of the lug. What do you think?