It can happen if a high power device is used continually. This is against the regulations, but sometimes people do things like connect a 3kW weather heater with a 13A plug, or connect several devices adding up to that power through a power strip.

If you run 13 amps through a cheap plug it will get very hot.

I've only seen this happen when the plug was fitted badly (pinched or damaged wires inside the plug) or someone use a nail in place of the fuse. People do stupid things like that all the time but it's not the fault of the standard, a fuse should blow if it's run over-current for too long.

Nobody else uses fuses.If someone does overload their circuit you have a fusebox for that exact reason.

It can when you overload a socket, which i have done accidentally. And it just melts the socket it doesn't set it on fire.