If the gauge of the wire is smaller than 12, say 14 which is very common for 15 amp circuits, then the wiring may go before the breaker will trip. The narrow gauge wire won't allow a large enough draw through the breaker itself to trip it, as all the resistance is elsewhere.
I have seen cases before where people have substituted higher amp breakers for lower ones. Outlets and things catch on fire without tripping the breaker.