My doorbell chime is partially broken; is this a wiring problem or a problem with the chime mechanism?
The simple path is to replace the chime.

The plungers get dirty and the added resistance makes them not slide correctly. 

Only one plunger should move for front, each end of it hits a different chime (Ding-Dong). For back, the other plunger is blocked so it only hits one of the chimes (Ding). 

It’s possible to clean plungers with alcohol then lubricate with graphite. Don’t use WD40 or oil, it attracts dirt and gets gummy over time, recreating the problem.