If you file a claim on two policies, one will typically be designated as "primary" and the other as "secondary". The secondary policy will likely offset (reduce) its payout by the amount you receive from the first.
Professionals like surgeons often "stack" a general group policy with a private "own-occupation" policy that pays out if they can't perform their specific job, even if they could technically work elsewhere.
Most insurers limit your total combined payout to roughly 60% to 80% of your pre-disability gross income.