Stray coupling to wires that aren't either powered or grounded.

A high impedance voltmeter is sensitive enough it will measure stuff off open wires acting as an antenna. There really is 25 volts there at that moment, but if you connect any sort of load it will go to zero volts because the stray coupling can't provide significant current to maintain a voltage across a load.

billhart-
Thanks for the input. I'm moderately knowledgeable about wiring -- enough to safely install and repair outlets and switches, run new wires, install breakers, etc. -- if I know what's wrong. What would you do next to chase this down?

Are these standard snap switches or master/slave dimmers?

They're standard 2-position snap switches.

billhart-
I just re-read your post, and I have another question. It is true that the two traveler wires to the other switch were disconnected at both ends when I got that voltage. So is it possible that this is not a problem, and that the reading is something induced rather than an indication of a physical problem with the wires/connections?

If both switches were disconnected or the lamp was out of its socket, then this is normal.

If the lamp was supposed to be connected through the other switch at the time, then there is something wrong because the lamp should have provided enough load to soak up the stray coupling.

When you rewired it did was the problem fixed? If not I would check the neutrals for a loose connection. That sounds like the culprit to me.

Brian-
I reconnected the wires, and I also checked all the wire nuts in the 2 boxes to make sure everything was tight (it was), and now the 3-way switches work properly, and the lights don't go off unexpectedly. I took the multimeter out and checked voltages, and I get the normal voltages through the switched circuuit, but I still seem to be picking up the 25 VAC indication off of one of the traveler terminals when the lights are on in the circuit. I also checked for current between the ground and that terminal, but got essentially nothing. And when all of the wires are disconnected and power is off, I get no stray readings. So...whaddya think?

Well if you are getting a reading off of the terminal (screw) from the traveler It maybe just picking up a reading from the common of the traveler in use. Maybe a cheap 3way. What kind or meter are you using also. You said you got a reading of "about 110 volts" which should be reading minimal of 118-120.5 or so volts in the US anyway. Im not sure about Canada. Are you in the states?

Well, the switch is a standard, residential-grade, 3-way made by Leviton (I mistakenly said Lutron brand initially). The reading is actually about 120 VAC...I was just being imprecise. And I am in the States. The meter is a fairly decent grade Micronta, bought at RadioShack many years ago (the batteries in it are fresh). It's not a digital readout: it's the typical swinging needle design.

When the light is on, one of the travelers is open at both ends. It acts as an antenna (capacitive coupling from its twin) and gives you the phantom voltage. As you demonstrated, it can't supply significant current.

All is normal.

billhart-
Thanks, again. That analysis makes sense to me. Sometimes I know just enough to be dangerous (to myself!).

This is ok for 3 way switch. Also you can get different readings, I guess depending on the dista between switches. All the houses I lived in hade the same thing. In my case voltage was from 25 to 40.