From my experience, if you already have a bank account, there should be no problem bearing in mind Willy's note above.
However, if you want to OPEN a bank account and then lodge the cash, you will have to satisfy the regulations around setting up a bank account before you can do so.
All that said, I'm not sure how the bank might respond to you presenting yourself looking like the guy from the TV programme Naked Camera...with a plastic bag full of cash etc..!!)