This is the question that quietly costs people the most money, and it is because the answer is two different answers wearing a trench coat.
Can you have an HSA while on Medicare? Yes. Absolutely. Keep it forever.
Can you contribute to an HSA while on Medicare? No. Not a dollar. Not a partial amount. Zero.
Those two sentences look similar and they are wildly different, and mixing them up is how retirees end up with excess contribution notices from the IRS.
The rule itself is simple. It is the timing that gets people, because Medicare has a habit of starting earlier than you think it did.
Let's untangle all of it. What happens to the account, what happens to contributions, the six month time machine hiding inside Part A, and the exact fix if you already messed this up.