Switch Wire Continuity

Checking the continuity of the wires between the switch and the chime is easy, finding the break (if there is one) is hard. To check the continuity...

1. Turn the breaker that supplies the doorbell transformer off, and verify power is off with a non-contact voltage tester.
2. Disconnect the wires in the chime that go to one of the switches (don't forget how they were connected, mark them if you have to).
3. Twist the wires you just disconnected together.
4. Go to the doorbell button and remove it from the wall.
5. Using a multimeter in continuity mode (or the lowest resistance setting), touch the leads to the switch terminals.

If the wires are good, you'll hear an audible tone and/or get a reading close to 0. If there is a break in the wire, you'll get a reading of infinity. If the wires are broken, inspect them along the entire length for damage. If you can't find the break or can't access the wires, replacement might be easier.

Switch Continuity

While you're checking continuity, it may be worth it to check the continuity of the switches themselves.

1. With the power to the transformer off, disconnect the switch from the circuit.
2. Measure the continuity in the default OPEN position.
3. Then press the button to close the switch, and take a reading in the CLOSED position.

When the switch is OPEN, you should get an infinite or very high reading. when the switch is CLOSED, you should get a low nearly 0 reading.

OK, this may be a side-effect of the way the Ring powers itself.

The Ring is in series with the chime. In series circuits, the same amount of current (amps) flows around the loop through both devices. And by the way, this problem crops up with Nest thermostats and dimmers too.

How does the Ring power itself? By deliberately letting a little bit of current flow through the circuit, including through the chime. The Ring (like the Nest and dimmers) is betting that a small enough current won't make the chime (furnace, lamp) activate. And it is guessing at how much current it can get away with.

So my guess is: you have a chime which is more efficient than most. Sure, when the visitor rings the bell, the Ring shorts the two wires to simulate a doorbell push, and then it lets go of that correctly. And then, it needs to recharge its internal battery - transmitting video over WiFi is hard work. So it allows current to flow at that lower rate. Unbeknownst to the Ring, that is enough to ring the chime. The Ring isn't timing out when it stops; it's just finished charging.

Now like I say, smart thermostats have the same problem: they charge by sending some current down the W line, hopefully not enough to engage the furnace relay. If that doesn't work, they give you an option to hook up the "C" line to power the thermostat directly. Unfortunately it does not appear the Ring has an option like that.

TLDR: Your chime is too efficient for the Ring to vampire-charge itself. Get rid of one.

Well, if you're in love with that chime, there might be another option, but it's definitely not UL-approved. You could alter your chime to be less efficient... by putting a resistor of appropriate size in parallel with the chime. Finding the appropriate size would take some measuring or experimentation. You would not want one which gets too hot, worst case someone holds down your doorbell button for a long time. A fellow shouldn't be able to burn your house down by holding the doorbell button.