The truth is some times banks run out of cash due demand on cash and they only give out the cash that has been deposited by some one else
So it normal for banks to be with out cash and this normal happens during the seasons like X mass, Ramadan etc where people are celebrating and everyone is trying to withdraw money from the bank
usually when someone need to withdraw huge amount of money they need to notice the bank before hand to ready the money before they can cash it out, it confirms your statement that bank usually just give back the money other people deposited for their liquidity so it does make sense that there's sometime no money in bank.
usually the bank themselves try to profit off the money deposited by investing it else where or give loan to people after all bank with their large scale of operation requires a lot of money if they don't do that i doubt there gonna be any bank office in every city.
there's reason why there exist term called Rush Money where people taking their money off banks due to mistrust and that usually causes bank to collapse.