If, on the other hand, you are talking about disability benefits through a private or employer-sponsored policy, the answer can be found in the policy itself. There is no one size fits all answer.
NY is one of the very few states which does have its own disability plan. You can find more info on it here http://ww3.nysif.com/~/link.aspx?_id=9E156FB25D114EA69C799B03F543858E&_z=z Note that this is only available for off work injuries and illnesses though handled through the same department as Workers Compensation.