I got overpaid on BAH for a while after a PCS move. After some time and I got a friendly but panicked, "You owe the U.S. Government $_____" email, with other correspondence to follow as required. I replied with a friendly, "I think I actually owe $_____ and I think you forgot to include 'save pay'. Other than that, I know I have to pay it back, I have a general idea how much you're allowed to take out each pay period, so let's get it paid back as quickly as possible." (Save pay is the thing where they can't lower your BAH if the rate drops after you move there; you keep the higher rate from the date you checked in.)
The rest of the conversation was also cordial beginning with, "I discussed it with my boss and you're right." The clerk felt really bad for me, since somebody in that office was responsible for forgetting to adjust it in the first place. I can't say I was too worried- I'd also noticed the error and it wasn't really my money.
Honestly, I felt a bit bad for the clerk for having to be the bearer of bad news. That's part of her job, but still... It was an impressive chunk of money.
I will say I found the whole experience quite amusing, but that's just me.