It turns out it is an extended billing number of the hospital's patients accounts, where I have an overdue bill for some years back. I should have told her that I don't like being automated-called and having to wait for someone to pick up.

Upon further investigation via Google, it seems the number belongs to MDB Communications, which is either an advertising company or a general contracting company ( I can't verify which). In no way does this number seem to be connected with the hospital, so it's either a scam or collections call.

Got a call at 11:12am from this number. A family member picked up and got an automated message telling her to please hold, told me the call was for me from Howard University Hospital, and handed me the phone while the music was still going. I heard another automated voice saying "To hear an important message, please press 2." After a minute or so, there was a noise like the hanging-up sound that happened two or three times. Since no one seemed to actually be talking, I hung up.

I called this number back and got an automated voice answering service for "Howard University Hospital Patient Accounts".  I stayed on the line until a human answered, but they weren't able to hear me so they hung up.
I'm assuming it's a collections office and they dialed the wrong number. Also assumed the line probably only allows outgoing calls, not callbacks.

Got phone call, no message.