On the face of it, what your policy covers is Bodily Injury or Property Damage arising from injury or damage which is both sudden and accidental, meaning that you did not intend for the damage or injury to occur. Although the policy won't say it, what is being insured here is your "negligence." Negligence, alas, is determined by the courts, but if there is little doubt of how the courts would rule, the insurance company intervenes to pay the injured party on your behalf.
Therefore, unless some specific exclusion applies (such as using the vehicle in the commission of a felony when the accident occurred) YOU'RE COVERED, period, end of paragraph.
My guess is that the denial may hinge on the "sudden and accidental" part. Thus, if the insurance company believes the conditon of the brakes was the result of your KNOWINGLY ignoring a problem with them, the accident may no longer meet the "sudden" test.
But that's a very thin argument and will not stand up in court. Other matters will need to come in to play. Were you drunk? I'm not saying that you were -I'm digging for some logical reason they might have to ask for their money back. Can you think of any circumstances that might have a bearing on it?
In any event, the company is going to be required to show good cause why you should repay them, if not to you, then to a court. You can't sue without a reason. Meanwhile, contact your agent or the claims department -in WRITING (fax is fine, but not email)- and pop the question. You may also call, but you need copies of written records and receipts of mailing in order to show that you made a good faith effort to communicate -and to do so politely -no threats or bad language, etc.
The comments you've received about "failed brakes" and rear-ending are all BUNKO. Front end, rear end, good brakes, bad brakes -doesn't matter unless you deliberately caused or contributed to these conditions.
So, short of an intentional act on your part, or ignoring a maintenance issue that you really, really should have taken care of, you're covered.
There is one more possibility: was the amount paid by the insurance company (to the injured party) MORE than the limit of your coverage? Say you carried $50,000 per accident, but the amount paid out was $60,000. If so, what the company is entitled to is the $10,000 difference. That's all black and white: you needed more coverage but didn't have it.
I did this stuff for years, so if you'd like to email me privately I'll be happy to answer.
For now, sit tight. They've only SAID they're going to sue.