I have been awarded Social Security disability benefits, but the insurer of my disability policy has denied my claim. Can the insurer do that?

Yes. The definition of disability generally differs between your policy and the requirements for Social Secuirty disability (SSD), and therefore you can meet the definition for SSD but not for your own policy. Many people think that SSD is more difficult to prove than LTD, but in reality they cannot be matched up based on requirements.