If my insurance company denies my claim, do I have any rights?

Absolutely yes. If an insurance company fails or refuses to honor its contract and pay a valid claim, you have the right to bring a civil action for damages against that insurance company. In addition to suing for a "breach of contract", you might be able to bring a tort claim seeking damages based upon the insurer's bad faith handling of the claim.

The amount you can recover varies based on the type of claim you bring and the jurisdiction in why you are pursuing your claim, but the good news is that you do have rights and can vindicate them.  In some cases there may be important deadlines or you may even have to arbitrate your claim, so be sure to talk to a lawyer about your rights as soon as possible.