Pursuant to NY law, in your situation, the terms of the policy control the right of the insurance company to terminate. Generally, if you have paid premiums for a certain period, you should be granted coverage for that period.
If the insurance company did not state that they had terminated the policy and were issuing a new policy, and you are still under the old policy, then you are entitled to coverage on the medical bills. On the other hand, if the insurance company terminated the policy and issued a new policy with a new coverage date, you are not covered for the missing time.
In either situation, I believe you have a claim against the insurance company for unfair practices. The first thing you need to do is request a certificate of coverage on your policy which shows the dates of coverage. This is a standard form that all insurance companies issue.
Next, you should check the policy information that they have sent you. Did the policy number change?
After you gather this information, if they tell you that you have the exact same policy as you did before the gap in payment, and your certificate of coverage states that you are covered, you should attempt to file the claim again pointing to your evidence of coverage.
If that is denied, then you should file a claim with the NY Department of Financial Services (the department in charge of insurance regualtions). It will be a "prompt payment complaint" and you can find the form for complaint at: http://www.dfs.ny.gov/consumer/fileacomplaint.htm
You will want to try to resolve this quickly before you medical provider sends you to collections and it affects your credit. They generally give you a decent amount of time before doing this, but you will want to get on the phone with the health insurance company and request the documents I referenced above ASAP.
If the medical provider attempts to send you to collections before the issue with the insurance carrier is resolved, it is wisest to pay the claim and seek reimbursement so as not to negatively affect your credit score.
If all that fails, I believe you have a claim for bad faith and breach of contract
against the insurance company. These types of claims are complex and have to be handled by an attorney. Hopefully, you won't get to that point and the Department of FS will force the insurance carrier to pay the claims.
Please let me know if you need clarification or further information.