I have Geico and filed one claim. They really came through okay but they are so dependent on independent insurance people that your local people are more important than the national outfit. It took about three weeks to totally settle a claim filed when my wife totalled her van which was reasonable.
I've been with GEICO since 1975 and have always had excellent claims service. Haven't had a claim in more than eight years now so I can't give and recent testimonials, though.