I think not, at least not very soon. People need paper money to buy stuff and pay daily expenses. Things may change in the future, of course, but right now people still rely on the banking system.
Actually, at the moment already exists many users who feel better using both payment systems, rather than using a single form of payment. As Bitcoin reaches more stability and its technology is recognized by more people, these people are likely to start using Bitcoin more often, as people tend to use what works more efficiently.