It depends on when you got your COBRA coverage.
COBRA is a federal law that gives you the right, to continue your health insurance once it ends because of job loss, divorce, death or other reasons. You must pay the full premium yourself.
If you already have COBRA coverage when you enroll in Medicare: Your COBRA will probably end. Contact your COBRA benefits manager to find out.
Caution: If you have COBRA and become Medicare-eligible, you should enroll in Part B immediately because you are not entitled to a Special Enrollment Period (SEP) when COBRA ends. Your spouse and dependents may keep COBRA for up to 36 months, regardless of whether you enroll in Medicare during that time.
If you become eligible for COBRA coverage after you are already enrolled in Medicare:
You must be allowed to take the COBRA coverage. It will always be secondary to Medicare (unless you have ESRD). You may wish to take COBRA if you have very high medical expenses and your COBRA plan offers you generous extra benefits, like prescription drug coverage.
If you have drug coverage through COBRA and you want to keep it, you should find out if that coverage is as good as or better than Medicare's drug coverage (creditable). You should have received a letter from the company providing your COBRA coverage letting you know whether your drug coverage is as good as Medicare's. If it is creditable, you may want to keep your COBRA coverage and delay enrolling in Medicare drug coverage. Many COBRA plans will not allow you to drop your drug coverage and keep your other medical coverage. Also, if you enroll in Medicare drug coverage later you will not have to pay a higher premium (premium penalty) as long as you join within 63 days of losing your creditable COBRA drug coverage. When COBRA coverage ends, you will have a Special Enrollment Period to enroll in a Medicare drug plan outside of the standard enrollment periods.
If your COBRA drug coverage is not creditable, you can switch to a Medicare private drug plan at any time. However you will have to pay a premium penalty if you did not enroll in the Medicare drug benefit when you were first eligible and have been without creditable coverage for more than 63 days. You will not have to pay a penalty if you can show you received inadequate information about whether your drug coverage was creditable.