I don't think this happens. Fiat currencies are controlled by central banks and governments and do not let this to happen. It will happen only if cryptocurrencies completely replace fiat money. The probability of this is very low.
I couldn't agree more. Bitcoin being use by a massive number of people will not reduce the value of the fiat currency for a reason that bitcoin does not replace it completely and for that the value of fiat currency stay the same. Also as what you have said, governments will not allow it to happen because fiat currency is still what they want and whay they control.