People turning 65 who are covered under employer insurance may defer Medicare enrollment until they stop actively working—or lose coverage from a spouse. You still probably should sign up for no-cost Part A, because for some insurers, Medicare hospital insurance becomes primary when you turn 65. COBRA is different. It is designed to bridge the gap between leaving a job and having another form of insurance—including Medicare. If you have COBRA, you have two reasons to enroll in Medicare at 65. First, COBRA often ends or becomes secondary once you’re eligible for Medicare. Second, if you miss your initial enrollment period into Medicare—the seven-month window around your 65th birthday—you won’t get another chance once your COBRA ends. That means you may pay a late enrollment penalty, which lasts as long as you have Medicare.