What are America's main/best or most popular banks?

PNC Harris Fifth/Third City National

As far as I understand, banks in USA have to be limited to operating within the state where they are incorporated. Of course the big banks have a subsidiary in every state, but they are actually separate banks. I have come across the situation where Big Name Bank A would not cash the traveller's cheques of Big Name Bank B (duh! what about the word "traveller" do they not understand? And the whole point of traveller's cheques is that they are GUARANTEED by the issuing bank)). And of course it was in a state where there were very few outlets for Big Bank B. And I have seen UK banks buy software developed for American banks, and then fall flat on their faces because the software could handle up to a couple of hundred thousand accounts, whereas the British bank had TWO MILLION. One of our building societies which used to be a household name has vanished because they made this mistake.