Door bells that do not ring could have transformer problems. Transformers convert 120-volt AC power into 12-volt DC power, thus regular voltage to low voltage. A doorbell transformer supplies the power to the button and when the button is depressed, the circuit is completed and the power flows to the bell or chime to cause it to ring.
Using a multimeter, you can check for voltage at the transformer. You should register 12 volts DC on the terminal end and 120 volts AC on the feeder lines. If both of these voltages check out, continue on to step four.