I am aware of banks that do this.
$3,000 is often the max allowed; the balance is paid with a Cashiers Check.
It is because the branches simply do not carry much cash.
If a customer insists on a large cash amount, they have to wait for it to be ordered for them, and then they must sign a form which reminds them of the risk of having that much cash and indemnifying the bank.
So yes, you can have it in your policy and decide your limit, with exceptions.
Year after year I come to back to this place - to remember the taste...!