Door bell button on Nutone system lights but when pressed goes out and no chime heard. What gives?

Somewhere on a wall is the chime itself. Lift off the cover and you will see a plunger rod with a spring, 2 if you have a front and back chime. I would guess that the rod(s) is gummed up. Have someone press the door button. Do the rods try to retract and hit the chime bar? If it is trying to move, clean them with a little WD 40 and put it back. If you do not see the rods move at all, the transformer could be bad. Honestly, a new replacement unit is easier to do then messing with an old one and they are $20 or less for a basic model.