First, a bit of clarification. There is no “Connecticut Division” of the BBB. The BBB is exactly as CapriRacer described, simply a (generally statewide) member organization to which the members pay a subscription and with a set of “bylaws” that the members swear to follow. It has no authority to resolve disputes.
However, many states have a Consumer Affairs Division of the State Attorney General’s office and they can investigate complaints, and if violations of law are discovered they can refer the matter to the AG for action up to and including criminal prosecution.
I had a new car many years ago that was involved in such an investigation. I recieved a letter asking me to bring the car in at my convenience (optional days were offered) for review of the work (allegedly) done. Mine was one of a number of cars that were reviewed that day. Representatives from the AG’s office oversaw the review by the dealer’s mechanic. I was told upon asking that it was part of an AG investigation. I received a full refund on the spot. I never did find out the details.