Progressive Slacking on Paying Accident Repairs

I gave my car to my sister (she has her own insurance on it), and she got rear-ended. Thankfully, no one was hurt, but the accident caused the exhaust pipe to fall down, and it’s been hanging by a thread ever since. The person who rear-ended her wasn’t the owner of the car. We got in contact with the car’s owner, who has Progressive, and submitted a claim through them.

Progressive sent someone out to take pictures, but they’ve been extremely slow to respond. It’s been almost four weeks now, and when we called, they even said they “weren’t sure if they were going to cover it.” How can this be? Isn’t it their responsibility to cover the damages? Is it common for this process to take so long?