Q: I have l large group cobra policy, and Medicare. Is Medicare secondary, as it was with the original policy and what if Cobra pays before Medicare, can they demand the money back from me? A: Once you are entitled to Medicare you are no longer allowed to continue on the employer ‘s group health plan. If you have a spouse or any dependants, they may take advantage of COBRA, but you will not. You are not able to have both Medicare and COBRA.