Insurance companies have an absolute duty to honor their policies and pay those claims their policies obligate them to pay. If they act in bad faith, they could be held liable for punitive damages in addition to the benefits they failed to pay.
That said, the terms, conditions, and definitions of disability in disability insurance policies vary from company to company, and from policy to policy. Your policy itself sets forth the definition and standards you must meet and the nature of the proof you must submit to establish disability.
Merely claiming that one cannot work and thus is disabled does not do it. Merely stating one has a serious disease does not do it. You must tie the disease or condition into an inability to do the level of work defined in the policy.
This is done by submitting adequate medical evidence -- your treating physician's reports, test results, specialist opinions, etc. While people can do this on their own, and insurance companies pay most claims quickly, in some cases the disability may not be very clear. It may be the evidence of a covered condition or illness is inadequate, or the causal relationship between the illness and the disability is vague, or the tie in between the condition and inability to actually work is missing.
In such circumstances lawyers help people establish the disability, by determining what additonal evidence is critical and assisting them in getting it --- benefits may have been denied because the doctor's report was not properly written to demonstrate the nature and extent of the disability and the causative factors, or the onset or likely duration of the disability. Social Security Disability lawyers do this all the time, and so do many insurance lawyers.
An insurance company's staff tends to pay much greater attention when lawyers get involved, as they recognize that the next step may be litigation (which nearly all insurance companies want to avoid) and in some cases, liability for bad faith by way of punitive damage claims.
You may also want to contact your state insurance department and enlist its help, but as most of them are overworked, understaffed and underfunded, it is far less likely to get you the results than a lawyer.
Finally, an insurance company is NOT bound by the determination that Social Security would make, nor vice versa.