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.
It is supported by government. But it's not "backed" by it unless you meant something else than what I'm thinking. When you say fiat is backed by gold, that means any fiat is equivalent to the amount of gold available and you can exchange your fiat to gold right away. So if you say that fiat is backed by government, can you exchange your fiat into a government? Definitely not.
~snip~
but in future we can expect that the government will start thinking about bitcoin to consider it as legal currency of the world.
With Japan and the Philippines (and other countries that I haven't read of lately) accepting bitcoins as a legal payment method to pay your bills, etc. I'm pretty sure other countries, especially the major ones (i.e. China, Russia, America), have already been considering accepting bitcoin as a legal currency.