We had a similar problem with a wired doorbell. There is a diode in the doorbell itself that causes the chime to ring fully. If the diode is backwards, you get just a small tone and can only hear the full chime if you hold the button down. If this sounds like your problem, open up the doorbell at the button end (or have an electrician do it) and reverse the diode.
Answers & Comments
We had a similar problem with a wired doorbell. There is a diode in the doorbell itself that causes the chime to ring fully. If the diode is backwards, you get just a small tone and can only hear the full chime if you hold the button down. If this sounds like your problem, open up the doorbell at the button end (or have an electrician do it) and reverse the diode.