I worked in a bank, so please, allow me to chime in.
We are a bank, yes, however, we are not the US Mint and do not keep millions of dollars on hand as some would like to think. We too have to protect our safety and if we are going to keep millions of dollars on hand, even a million, we are risking our lives. Further, we try to order what we know we use weekly. There were times we would get non account holders coming in to cash large checks because their bank would only allow them to deposit the check if a hold was placed on it (Reg CC). This is either because their account was new or they do not know how to handle their account, so they would have to wait for the funds to clear. We would also get account holders that come in and need to w/d 20K-30K, which takes a lot of the money out of the bank.
Also, one thing I was taught years ago when I was a teller...if I am short and I know who I gave the excess money to, I was instructed to call that customer and say, "Mr. Jones, when you came in to the bank, I inadvertantly gave you too much money. I will need for you to bring that money in by x date." So, they may not be certain it is the OPs money, but hope s/he will just go ahead and bring in the money and call it a day!