[Varley]

An urban supply (in principle, underground cables) may have (has usually) grown by extending the distance of a consumer from its substation often by splicing-on more of the same. It can (does) occur that the impedance at the newly more distant end is too high for the substation to deliver sufficient current to lift the substation protection should a distant fault occur. Then, indeed, the cable will burn out instead.

As explained to me by interesting network engineer after just such had happened to me (a line to neutral fault putting 400 V or so across some of the domestic kit, including two of the utility's meters, none of which worked terribly well afterwards). Fault was within a metre or so of the utility fuses but on the incoming side of them. One line conductor eventually found open and a few metres of new cable spliced in to replace damaged. Quite fun following the process (although rather dark and cold interim), modern technology - reflectometer - put fault possibly at edge of lawn or McDonald's car park with appropriate excavations made to confirm (so closing McDonald's car park, what a shame!).

Ancient linesman maintained that fault most likely in 'turk's head' termination casting from which individual tails exit to fuses/meters. And so it was.

Overload rather than short circuit is different. Proper electricity does not usually use fuses to protect kit or its local wiring from overloads, that being the duty of some sort of protection relay. Fuses serve to disconnect a carbonised theatre from the mains so as to allow a continuing supply to the neighbours - even if that includes a McDonalds - which would be lost until said carbonised theatre had been otherwise disconnected from it.