Not unknown for a 13 amp fuse to get 'tired' and fail for no reason other than that ..but not usual. A 13 amp fuse won't usually fail without a reason. Do you know the power rating of the dishwasher? I read they are typically around 1500 watts but yours may be different. If you have the instruction book for the dishwasher, the power consumption figure should be given somewhere.
If it is 1500 watts (1.5kW), a 5 amp fuse will fail anyway because the current taken by the dishwasher works out at 6.5 amps (current = watts ÷ voltage). A 5 amp fuse might last a while but I'd expect it to fail without a fault being present. So using a fuse of that value wouldn't really be a test of whether a 1500 watt dishwasher is faulty my opinion.
However, if your dishwasher is only 1000 watts (1kW) then a 5 amp fuse would prove the point since, under normal operation, the current requirement would work out at 4.3 amps. The fuse would get tired and fail eventually but should see the dishwasher through a cycle or two. The test would hold up for a dishwasher of that rating. So it does all depend on the normal power consumption of the dishwasher.
The colour of the failed 13 amp fuse may not be relevant. The ends can become discoloured after a while. Difficult to say without seeing it. You could show this post to your electrician for an opinion. I can't recommend anything for safety reasons but he may advise trying a 10 amp fuse or even another 13 amp one. The 13 amp fuse is there to blow if something goes wrong and it seems it did do its job (if a fault is present). You are also covered by the house cut-out.
My advice would be to give the electrician another call, preferably stating the power rating of the dishwasher. If it is more than a 1000 watts ask if the 5 amp fuse test would prove the presence of a fault. If a 13 amp one blew, I'd say there's a 75% chance that there is a fault ..but I could be wrong.
Also, consider taking the fuse to the electrician and get him to measure continuity through it. It might just be OK. What can happen sometimes is that the mains lead becomes intermittent internally where it goes into the plug, especially if the plug is plugged in and out of the wall socket a lot.
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