I'm used to scams where they ask you for moneygrams, but not the other way around.
I applied for an apartment Oct 7th, and moved in Nov 13th. When I tried to pay December's rent, I saw that my account with the apartment complex lists some 4.5k in un-applied rent credit, a little less than three times the monthly rent. According to the front office, that was comprised of 5 separate moneygram payments (four for 1k, one for ~$500) made for my apartment Nov 4th, I assume after the previous tenant had vacated. Those payments weren't listed when I first moved in. The front office told me not to worry about it, and suggested a grandparent or someone might have made those payments - not realizing that they had been made before the start of my lease (and unlikely regardless). I paid December's rent anyway.
The payments were by physical moneygram checks, so I assume they can't be rescinded, and also as far as I know wouldn't have been some automated system. They also were made to the apartment complex for my current apartment, not to me directly in any sense, so I'm assuming I don't have any reason to trust that they'll stay applied to my account. I imagine it's a mistake of some kind, rather than something nefarious, but I don't know.
I was given contact info for someone to escalate the issue to, but is there anything to be concerned about if they come back and tell me to treat it as simple good fortune?
Best case is some stranger wrote a check to a random address for a laugh, which is more than slightly doubtful. Worst case I can think of is I rely on the money and they later tell me, "yeah that was a mistake sorry, so now you're three months late on payments", and the person who accidentally wrote 'apt 1#' instead of 'apt 2#' on their check is out the likely meaningful sum of 4.5k.