Big companies / banks may come around to using Bitcoin when it becomes successful and attains widespread adoption.
What would banks be using Bitcoin for? It's a complete and total eclipse of their business model; there are no banks in Bitcoin (hence "be your own bank")
For settlement between them ultimately is what I think Andreas A. speaks about. AOL and other closed networks would still need to communicate between themselves using a standard protocol (if they ever wanted to allow more freedom for their users), and when that protocol is open to anyone, people just skip the private networks and use the Internet directly.
But you are correct, it is the opposite of the banking business model. It would be out of necessity at first and then later banks would be more like BitGo or something, offering security, insurance, etc. IMO
