If you had an accident after getting the quote but before taking up their offer, it is reasonable to expect the company to increase the cost; there is normally a caveat in quotes to that effect. As I said, the No Claims Protection you paid for with your original company only applies if you continue to insure with them. Even if you had stayed with your original company your premium would have gone up as you had become a greater risk, although your no-claims discount would not have been reduced.
When you get a quote you are not asked what level of no-claims bonus you are on, you are asked how long since you had an accident/claim and that is what determines the premium you will be asked for.