One circuit of lights comes on dim when another is switched on
As mentioned in my comment, I would say this is caused by phantom voltage or induced voltage.  This is a problem that is seem more these days because of an LEDs ability to run on very, very low power levels.

What happens is that the alternating current in one wire that is running parallel and very close to another wire will induce a voltage in the wire that is currently switched off.  When all the dimmers are off, there is no current in any of the wires, but when one is switched on and the electricity starts to flow, it causes a voltage to appear on the nearby circuits.

This is not enough to shock you, but it is enough for non-contact voltage detectors to pick up, and it can often be measured with some multimeters.  Other lighting technologies need way to much power for this to affect them, but the modern LED can have a small glow with the tiny amount of current.

I also found this question where they were able to use a resistor wired parallel to the switch to stop glowing, but they were having an issue caused by a switch with an indicator lamp and not induced voltage, so I'm not sure if that would help in your situation.