I did have one difficult claim with Peter Best, but that was 24 years ago so it might be unfair to hold that against them still!
I was unfortunate enough to have an accident in my MG BGT - I was stationary in a street, waiting to pull in to a parking space, when a Corsa pizza delivery van drove into the back of my car. Front of the Corsa looked wrecked, rear of the BGT had a broken tail light

. But structurally the body was bent underneath (evidenced by a crease in the rear wheelarch, and the bootlid didn't close quite so well) so the car was written off.
The issue was that I'd taken photos and sent the Agreed Valuation form off only a day or two before the accident, and it hadn't been processed and approved yet.
The Agreed Valuation department went ahead with their thing (probably unaware of the claim), and duly agreed the value I'd asked for. Meanwhile the claims department did their thing, and looked for the Agreed Valuation certificate, found it, and said that since it was dated after the accident it wasn't valid, and they were only going to give me what they wanted to give me, which was about 2/3 the value.
I pointed out that, even if the certificate wasn't valid, they themselves had still said it was worth what I was claiming, so that's what they should pay. They then claimed that the photos appeared to be old and not representative of the state of the car, which was bollox (apart from the photos not showing the accident damage!).
Eventually they conceded, and paid me what I wanted. Just shows that even specialists try the standard tactic of initially trying to get away with a low settlement, in the hope that you'll just take it (and I bet plenty do), but if you persist you can get a fair settlement - not the only time I've been through that cycle!