All 4 of mine are covered by Sainsburys. I've only had to claim for one of the dogs but claimed twice on him with absolutely no quibbles and received money quickly. Each subsequent dog on the policy has been at a lower premium too so pro-ratering it out, it works out pretty reasonable too.
I have all mine covered with them after M&S rocketed. Youngster had a knee op followed a few months later by a sock removal with complications with claims totalling about £2.5K and his premium went down this year!!! Money always received promptly.