Doorbell button wiring no voltage?
Generally the chime will have a solenoid that gets pulled or pushed when the circuit is completed. The solenoid bangs against the metal and you hear the nice chime. [I thought these were 24V circuits - so be sure you check your door bell system out and verify that first.

Check your voltage on the solenoid - you can check resistance with no power attached (should be some low ohms around 4 ohms probably.)
When your door bell button is pushed voltage will flow from the Transformer into your solenoid - so checking the voltage at the solenoid will provide an answer as to what is working and what is not working. 

You should be able to measure your 'voltage drop' at the solenoid when the doorbell circuit is made. Use Ohms Law - the voltage you know is 16V, the ohms is the resistance of the solenoid. Your solenoid even though it is an inductor has resistance - so it is your equivalent to a resistor in the example link.

Here is a quick link:
http://www.wikihow.com/Calculate-Voltage-Across-a-Resistor

Some things to keep in mind:
1: Solenoid sticking - clean it and use graphite lock type lubricant.
2: Solenoid open or shorted..
3: Transformer not working or "Overloaded" - too much current draw causes the voltage on the transformer to drop and it will not be able to pull in the solenoid. This indicates either something shorted - or a wrong doorbell chime, or a transformer that is simply worn out.
4: Doorbell switch bad .. or too much resistance - simply put one wire to the other and see if the thing works - if it does the switch is bad.