It depends upon what you signed with your dentist. If they agreed to bill insurance and did not, you should only be liable for the uninsured portion of the services. However, many agreements say you are responsible for full amount of bill and insurance will reimburse you what you paid. Look at the agreement in your dental file.
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You probably signed an agreement with this dental office that you'd be responsible for all billed charges whether covered by insurance or not, and that they'd bill your insurer for you as a courtesy but weren't obligated to do so. So on one hand you're obligated to pay for the services rendered to you.
But on the other hand, once they undertook to bill your insurer for you, they're under a duty to do it properly. if they screwed up the bill, and it wasn't paid because of their incompetence, then they're negligent, and you have a cross-complaint against them.
Your query states "something wrong with coverage," and isn't clear on why the claim was rejected - was it submitted too late? Their fault. Was the work not a covered benefit under your policy? Then not their responsibility. The answers to whether the claim would have been covered had they handled it properly determine whether you owe the part not covered had it been billed properly. You're certainly responsible for your deductible payment.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.