To answer the question of what the maximum fault current attained during a short circuit just before a fuse interrupts the circuit is called, we need to examine the given options:

1. Rated current: This is the maximum current a component can carry continuously under specified conditions without degrading its performance. It is not related to current during fault conditions.
2. Melting current: This is the current level at which the material of the fuse begins to melt due to heat. While related to the operation of fuses, it is not specifically about the fault current.
3. Breaking capacity: This is the maximum current a breaker can interrupt without being damaged. It's related to circuit breakers rather than fuses.
4. Cut-off current: The maximum current reached during a fault before the fuse opens the circuit, effectively limiting the fault current. This is the correct term for the scenario described.

The correct answer is Cut-off current as it is specifically defined as the maximum fault current present just before the fuse breaks the circuit. Fuses are designed to melt and interrupt the circuit when this current reaches a dangerous level, preventing damage to other circuit elements.

In power systems, it’s important to know these parameters to ensure protection devices operate correctly during fault conditions. Understanding fault currents and their implication on system protection is crucial for designing safe electrical systems.