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You can delay enrollment in Medicare if you are covered by your or your spouse's employer's group health insurance plan if the employer has 20 or more employees. If the employer has fewer than 20 employees, you should sign up for Part A and Part B when you're first eligible, as Medicare will pay before your other coverage. When your employment or coverage ends, you have eight months to sign up for Medicare to avoid a late-enrollment penalty. If you have COBRA coverage, you will not be protected from late penalties if you wait to enroll in Medicare past the deadline.