Is "James Bond" just a title?

Two buddies of mine have insisted that they've seen a deleted scene from a Bond movie (possibly "Living Daylights") in which James Bond explains to someone that the name "James Bond" is handed down to the top MI-6 agent whenever the current person using the name dies or retires. They do this in order to maintain the illusion of a British super-spy who can never be beaten. I called "BS" on this, but they're both sticking to the story. Can anyone here back this up?