Most insurance renewal documents state how many years no claim bonus they include and that was the evidence I used when I switched from DirectLine to LV recently. If the renewal doesn't make it clear and you're looking to switch then you can ask your existing company to provide your no claim entitlement in writing. If you get a quote from One Call's web site, the available answers to the no claim bonus question only goes up to 10 so 35 years is a weird demand.
With premiums ever rising, more and more claims being made and stories about companies using any excuse to get out of a claim, long ago I decided to stick to main-stream well known insurance companies.
An expensive lesson but next time remember, Google is your friend
P.S. Check your renewal documents, I left DirectLine because not only did the base premium go up, it included an increase on the voluntary excess and they added an excess for windscreen repair claims. LV were a very similar price but I do a lot of business miles and LV just have a >25,000 business miles option whereas DirectLine's business mile options (and premium costs) went up in blocks of 5,000 to 100,000.
I'd also stay away from insurance comparison sites, you'll never hear the end of them and they are more interested in putting you on to a better commission deal for themselves than they are getting you a good deal. After quote Swinton online as the best deal, £200 more than DirectLine, they phoned me up within minutes of completing the online form and said Swinton really wanted to talk to me and transferred me straight to Swinton. When Swinton answered they said they thought it was a great price, I said they thought wrong and hung up. 5 more calls later they finally got the message to f'k off. Great TV adverts though
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#TooMuchTimeOnMyHands