Two possibilities:

According to James Rickards, gold is going to be worth $10,000 an ounce of gold (today USD 1.3k). If Bitcoin reaches $10,000, I'd say the catastrophe would be even bigger. Fiat almost worthless.

OR

The financial bubble we are experiencing today gets so big that bitcoin 16-folds its current price, and the same happens to every other asset (not necessarily in the same magnitude. Fiat almost worthless.

Final result: Yes, fiat would be worthless.