There is a possibility but I can say it is very slim. People will always use the currency issued by their government. No matter how good Bitcoin is, people will still believe in the currency that is supported and endorsed by the authority. Even if Bitcoin had been approved and acknowledge as currency, I still doubt that it will replace fiat.
But isn't bitcoins already endorsed and supported by the authority once it has been approved and acknowledged as a valid/legal currency? Because that's actually the point why did the government approved it in the first place.
With all the international or long distance transactions going on every single day, I believe that bitcoins will take over most especially when the government has approved bitcoin as currency. Even without the government's approval, bitcoin still has a chance to take over because of the same reason (people are sending money overseas/long distance more and more).
Fiat money is a big power backed by government, bitcoin simply cannot replace it instead of become an alternative currency which used for send/receive money overseas, useful only if you send it pass through your country to another which use bank service or remittance require higher fees and some documents to fill. If I want to send money domestically inside my country, using bank service is a better idea and it's free or cheaper than bitcoin.