gary1957 said:
does something arrive in the mail, or am I supposed to go do something
You have to go do something, Gary. The benefits officer is right and this is the single most common Medicare surprise for people who delay claiming.
The rule, generally: enrollment in Parts A and B is automatic only if you are already receiving Social Security benefits when you reach 65. If you are not, nothing happens on its own and you sign up yourself. Your window is the Initial Enrollment Period: seven months total, starting three months before the month you turn 65, including that month, and running three months after. Signing up in the three months before your birthday month is what gets coverage starting the month you turn 65, so treat the early end of the window as the real deadline rather than the late end.
What it costs to miss it, since that's what's worrying you: the Part B late enrollment penalty is 10% of the standard premium for each full 12-month period you could have had Part B and didn't, and it stays on your premium for as long as you have Part B, not for a year or two. Part D has a smaller ongoing surcharge on the same logic. Those penalties are the reason this is worth a calendar reminder rather than a mental note.
Three exceptions and complications worth naming precisely. First, if you or a spouse are still working with active employer group coverage, you can usually delay Part B without penalty and enroll later through a Special Enrollment Period. Retiree coverage and COBRA are NOT active employer coverage for this purpose, which catches people out badly. Second, if you have a health savings account, Medicare enrollment ends your ability to make new HSA contributions, and Part A can be backdated up to six months when you enroll after 65, so contributions made in that backdated stretch become a problem. Third, since you won't have a Social Security check to deduct the premium from, Part B will be billed to you directly, usually quarterly, until you claim. The site's Medicare explained walks through the parts and windows, and when to claim Social Security covers why the two ages no longer arrive together.
Your own dates and coverage history are what decide which of those applies to you, and Medicare.gov and SSA.gov will both answer for your specific record, free. I can't do that from here.