The simple answer is no. Each bank has its own requirements and restrictions on cashing checks. However, there are a lot of options for cashing checks.
Whatever bank the check is written from is going to be able to cash the check. Regardless of what bank the check was written from, most banks will cash checks for any of their own customers. There may be a fee applied to cashing checks at any bank without customer status of the writer or recipient. There are also many companies that charge a nominal fee to perform check-cashing services for a check from any bank.