Hello all. Our house is approximately 13 years old. Each morning when both the iron and blow dryer is plugged into two separate GFI outlets in the master bathroom, the circuit will trip the 15 amp breaker in the breaker panel.

Is it possible to upgrade the breaker to a 20 amp without overloading the existing wire? (It's rare that the blow dryer and iron are on at the same time, but when we don't think about it and plug them both in - trip)

Am I mistaken to believe that the circuit should be tripping at the GFI outlet instead of the breaker at the panel? Is it possible that my GFI outlet is faulty and could just use replacing even though it tests fine using its own test button?

Any light you could shed on this is greatly appreciated.

The GFI is to trip if there is a difference of current being drawn from the hot and neutral.
It's probabbly OK
If you have #12 wire, you can increase to a 20 amp breaker. Is their a circuit on the opposite side of the wall you can get to same bath, could have 2 circuits, then neither would blow.
Needs to be wired so both can be on at the same time, or use 1 at a time.

A GFI is NOT a circuit breaker, so it will not trip on an overload.

Your situation is common which is why they changed the code to require 20A circuits in bathrooms.

Other than running a new circuit to one of the rooms you have to live with the occasional tripping.