I think that this will never happen. The dollar is the currency that the government supports. That's why the dollar will always exist. The government can not trust Bitcoin.
Government support is not the reason why this can never happen. The main reason why this can't be happen is that, bitcoin is just a digital currency which replaces only fiat money online. Hence, at the end of the day, it has no value if it can't be traded with fiat money since fiat money is only the official money which all over the world accept wether it is small or large community.