Breakers (and fuses) have two ratings
The continuous load rating (whats printed on the handle) and the interrupting current rating
which is printed on a label on the side of the breaker.
The higher the breaker rating, the higher the interrupting rating typically.
So if you overload the circuit above the breaker rating but below the interrupting capacity
of the main breaker , you will trip the lower circuit
If the "short is higher than the circuit breaker interrupting rating, the main breaker will trip
to protect the circuit and the the actual lower interrupting rated breaker from damage
So if you have a dead short , it may trip the main before the circuit.
Or a sub-panel breaker before the breaker on your power bar.