At the present time, this is unlikely to happen, but it's possible to realize that people are getting tired of inflation, there is a high cost in using a currency that loses value at a high rate, year after year. Maybe Bitcoin will not make people stop using fiat currency, but at some point when they realize the benefits of using Bitcoin, the number of fiat currency users could drop significantly.
As long as people are having appropriate reasons to switch over to bitcoin payment system, how we could expect the survival of fiat systems ?
Other than economical reasons, online usage like on the go with the mobile phone kind of requirements will push bitcoin to go mainstream of payments system. By considering all other potential benefits of bitcoin along with this online usage, I believe bitcoin will replace fiats slowly.