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I've just had an unusual one with a building that's had a cu changed. The electrician who did the change found shared neutrals so just disconnected them and told the client, can't fix that and left. There were also issues where sensors for bathroom lights had been wired to the cpc (which he'd removed because that's illegal) as it was switch line only and that was mixed up with one of the bathrooms having been sharing a neutral with hallway lights so no power at all. Or should I say, you test the wires you get voltage but no neutral so it disappears under load.
Best part of it was I had a very strict deadline and the client couldn't tell me anything about the previous electricians or work done as it had been quite a while and the house had been empty so he'd not been on top of the work. So I had to figure it all out from scratch.
But getting back to the light problem, one led on in a circuit when the dimmer is switched off has very few problems that could be related to missing neutrals or damaged wiring. If all the leds were on very dimly after being switched off that would be a different story. I'll be interested to find out if it is anything else.
One way to test is put halogens in the circuit.
Sorry rushing responses. I should have said wiring faults/issues can generally a bitch to identify.
I'm in the middle of looking at wiring problems in a non domestic installation.
Funky neutral/ground setups and nails through wires have been common issues I've seen for circuit issues.