I have a 3 way circuit providing current to 4 recessed LED lights. When the switches are off, the LED lights still burn very dimly. Approximately 23 volts in one of the fixtures.
What causes this?
Any pair of plates or wires next to each other will have a parasitic capacitance between them. For a typical length of 14 AWG NM cable between two switches, we're talking about roughly 75pF to the meter between adjacent wires, such as the two travelers in a three-way switch.
As a result of this capacitance and the fact that mains electricity uses AC not DC, a tiny amount of current can flow through a three-way switch setup when the switches are configured to turn the load off. For say a 10m cable length (750pF of capacitance) on 60Hz mains, we get an inter-traveler impedance of 3.5MΩ and an allowable current flow of 34µA with the load shorted.
This is an insignificant amount of current if you're an incandescent lightbulb; however, LEDs themselves require much less current to run, and the driver circuit in your LED fixtures is clearly finding enough power there to sustain operation at a low level. (Some drivers have far larger capacitors in them, so they charge up, try to start up, drain out the large capacitor, shut down again, and then start charging again, resulting in flickering behavior instead of the dim glow you see.)
The 23V reading is similar -- a typical modern voltmeter has a very high input impedance (on the order of 10MΩ), and is going to read the parasitic voltage that was coupled across as a result. (If your meter has a LoZ function, switching to that will cause the parasitic, or "ghost", voltage to go away.)
Some LED light fixtures don't work with some smart switches and they don't shut off entirely.
I just had this exact problem with Lutron dimming switches and some cheaper LED lights I bought.
Check the manufacturer's recommendations for which lights work with their switches.
There's not much chance for magnetic coupling because hardly any current flows when the switches are off -- I suspect that you may be seeing capacitive coupling between the travelers.
This, together with the natural autoinduction always present in house wiring, has formed an accidental oscillator with a frequency that is some exact integer multiple of 60 Hz, so it is driven, or pumped, by the line voltage.
The amplitude of the oscillator current is just barely enough to trigger the LED drivers and cause a visible flicker. Any disturbance of the wiring should change the oscillator frequency enough so that it is out of tune with the 60 Hz line voltage, causing the phenomenon to disappear.