Will wet string conduct electricity

Like enough current to light a light bulb?

That's extremely unlikely. If you soaked the string in something that conducts well - such as a brine or something like that - then MAYBE.

Probably not, but it depends on the voltages and currents involved. And on the impurities in the water, as that determines the resistance.

I can think of conditions where the bulb would light up. A short string wet with salt solution, and a sensitive LED light. But there are many many other conditions where you get no light.

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Yes it can.
You can get shocked through wet string.