A small appliance develops an internal fault, and starts to draw more current than it would normally, sufficient to blow a 3A fuse, (but not a 13A) before any damage is done.
But, if the appliance is fitted with a 13A fuse and 13A rated cable, then the fuse won't blow, the cable won't be damaged, but the appliance catches fire because of the heat generated by the excess current flowing through its internal components not designed for such.
Central heating systems are also fitted with 3A fuses in their FCU, yet the cables used to wire them could be capable of carrying 13A. However the programmers, PCB's etc., could all be damaged by too much current. For example relay contacts often weld together. A 3A fuse stops the current flow before any damage is done.
But, if the appliance is fitted with a 13A fuse and 13A rated cable, then the fuse won't blow, the cable won't be damaged, but the appliance catches fire because of the heat generated by the excess current flowing through its internal components not designed for such.
Central heating systems are also fitted with 3A fuses in their FCU, yet the cables used to wire them could be capable of carrying 13A. However the programmers, PCB's etc., could all be damaged by too much current. For example relay contacts often weld together. A 3A fuse stops the current flow before any damage is done.