Cage bird not molting

Hi
Any body out there have any ideas on this.
Have two finches seperate cages, one started to molt about a month ago.
the other one has shown no sign of losing any feathers, he has stopped singing as I expected but not even one feather on cage floor. he is on a varied diet with greens a couple of times a week plus cuttle bone, he has a bath every day which he really enjoys, looks outwardly healthy. The reason that I am concerned is that a number of people inthe bird scene have told me that this is a rare occurence and invariably the bird declines and dies.
Any advice or is this an old wives tale.
Thanks