If the payee on a check is only the bank, can any person bring it into the bank and cash it?

Anyone could certainly present (not cash) the item, but only an officer of the bank may actually endorse it, on behalf of the bank, back over to the presenter for encashment. The bank officer has a certain obligation to ensure that the bank has no interest in the check prior to making that endorsement. Based on the way the question was asked, that's really the best answer I can give - hope that helps!

Banks are (and should be) wary of checks payable to the bank. Most banks will accept them only if they are for payment of money due the bank, or if the checks are clearly captioned to indicate their purpose. For example, a bank would certainly accept such a check in payment for safe deposit fees, loan payments, credit card accounts, and federal tax deposits (does anyone do those any more?). But it would be questionable or inadvisable for a bank to accept such a check for credit to someone's account, except in limited situations.

John,
Wehn you say that "would be questionable or inadvisable for a bank to accept such a check for credit to someone's account" you mean deposit account, correct?

Also when checks come in payable to the bank, is there a best practice for an endorsement? Should the officer that accepts it initial the back? Should a credit stamp be used?

That's correct. Suppose ABC Builders sends its accountant in with a check payable to the bank that's supposed to be for a tax deposit (TT&L payment). Instead, the accountant hands you the check with a deposit ticket to the accountant's own account! Don't think this hasn't happened. I've seen it close up and dirty. To the cumulative total of $300,000 before ABC Builders heard from the IRS about failure to deposit taxes. Guess who had to pay those taxes?

If a check is properly payable to the bank (for its own accounts), the normal proof department endorsement is probably fine. Often, departments have stamps they use to indicate they processed such checks so the bank knows what to do if one of them bounces.

Thanks John.