Interesting phenomenon, I have some LED corn lightbulbs (from DX), which have 48 surface mount LEDs.  Two of these are in a hallway set up with 3 different light switches.

When one of the switches is in a particular position, and I turn the lights off using either of the other switches, most of the LEDs stay dimly lit on each of these bulbs, indicating that there is a low voltage being fed to them still, but not enough to light them all fully.  Other non-LED bulbs on the same circuit do not show any sign of life.

If I turn that particular switch to the other position, then switch the lights off on the other switch, then these lights will go dead.

I haven't worked out the circuit required to have 3 different switches controlling the lights, but I am guessing that one of the lines most likely runs close to the power supply for my hot water, and could be getting a small charge by induction.  Is that likely?  What other causes could there be for this charge on one part of the lighting circuit?

Should I get this investigated by a professional or ignore it?

And before you ask, no I haven't yet done the logical obvious test of turning off the hot water cylinder to see if the lights then go dark.  Must get off my butt and try that sometime.

[Update:  I switched of the hot water and it made no change, so the charge is coming from elsewhere]