Etiquette books in the United States used to tell people that this was what was supposed to happen.
However, in actuality this almost never happens, because it is silly advice. It would cause a great deal of confusion, especially in cases where the suffix was placed on the birth certificate and so can be considered part of the legal name. It's certainly not what happened with John McCain, for instance. He is still JohnSidney McCain III even though his father and grandfather are both long dead.