[Maintainer: Please note that the rec.guns FAQ disclaimer, as well as the disclaimer contained in the last paragraph of this document, apply fully to this article.]
I do not claim to be the best legal researcher, but I just completed a search of some federal law. We must remember that the banks are privately owned, but they are Federally insured, and as you know robbing a grocery store is a state crime but robbing a bank is a federal offense. I found nothing to indicate that mere possession of a firearm is illegal. I found several laws that make possession of a firearm an offense when the intent to commit a crime is also involved or where an actual other crime is committed. That is to say, if a person carries a firearm into a bank with the intent to commit a crime therein, then he is guilty of an offense. Additionally, if a person commits a crime, with or without the gun, and he possesses a gun an additional offense has been committed and additional (severe) penalties can attach. Therefore, I figure as long as a person merely carries a gun and does not brandish it, have the intent to use a crime without it, have the intent to commit a crime with it, or actually commit a crime while in possession of the gun, he is safe.
Though I must now add the obligatory legal disclaimer. I'm not a lawyer yet, and this shouldn't be construed as legal advice approving of the carrying of guns into banks. Though to tell you the truth, I've carried into banks in AZ, open and concealed. They never seem to have a problem.