From my view-point of a power system analyst, I see the following:
In industrial environments (LV and MV systems), the higher short-circuit level advantages are: less voltage drops due to transient loads (ex: motor starting across the line); less impact of harmonic loads (harmonic currents cannot cause too much variation on the harmonic voltage distortion where the short-circuit level is expected to be higher); less over voltages due to bolted ground faults. Disadvantages: higher incident energy due to arc faults in the switchgears; higher probability of devastating burning failures in motors and general equipment due to ground faults specially in solidly grounded neutral systems; a lack of an effective ground protection which is simple and cheap in solidly-grounded systems with high levels of short-circuit currents usually found in industrial environments.
In distribution lines (MV and HV): only high short-circuit levels are desirable for the overcurrent protection to trip and if the distribution system from the power company feeds an industrial substation, this higher ground fault level is responsible for the higher (earth) grid current which causes higher touch and step voltages in the grid and in the entire industrial grid if the grids (the main substation grid and the industrial grids) are interconnected, as it is usually the case.
In brief, short circuit levels should be large enough just to be detected and could be specified through various grounding techniques at the neutral point of the source. Usually, the system voltage plays a major role in such calculations. The main limit is related to power cables thermal withstanding capabilities. Such limits are known to the protection engineer during protection design.