^^^ Unless the cause of the accident was directly attributed to the cause of the accident, the insurance company doesn't have a leg to stand on, literally.
This same question was asked by a number of insurance carriers to various tire manufacturers - what happens if you run excessively high air pressures in a tire. As I was actually involved in one of those tests, I can say that you'd have to run stupid high pressures to grossly effect the traction on tires, under most operating conditions. I'm talking about doubling the max cold tire inflation pressure on a tire in some cases. Also depends on the tire, some tires failed right away, some took a significantly long amount of time before they failed - at those excessively high tire pressures.
Running 5-10PSI above the rated OEM tire pressures - IMHO, no big deal. You might hear and feel more road noise, but handling, braking, and acceleration will not be grossly affected. Some situations, like standing water on the road - you'd actually resist hydroplaning with a tire plumped up higher. With loose surfaces, like lightly packed snow and sand - that higher tire pressure will work against you.
I look at it as a built-in safety margin - a tire tends to run cooler (tread temperature) with more air than less. Not everybody checks tires pressures on a regular basis, and just visually looking at a tire will not indicate how much air is in it (ie, a tire with 35PSI looks more or less like a tire with only 20PSI).
Remember the Ford Explorer / Firestone tire issues in 1990? Ford at the time recommended running 26PSI in the tires - to give the car a more agreeable ride. Problem, at that particular pressure, the OEM tires fitted to those Explorers were grossly underspec'd - ie, they couldn't hold up to the weight of the truck. Couple that with owners not keeping an eye on tire pressures and you got a recipe for all those accidents, tread separation issues. Where the tire ran so hot, it literally peeled apart under the strain. Other SUVs are the time were running the exact same tire but at 32-35 PSI - they didn't see issues related to tread separation / overheated tires.
