I’m posting this tip because it took me a course of a week to enter the right combination of search terms to get an answer.
Symptom: On OS X, your voicemail stops working. They are listed in your history, but when you click to play, the playback window pops up for a brief second and then vanishes, never playing the message.
Cause: For some reason, Skype relies on your having a microphone attached before it allows you to play back messages.
Solution: Plug in a microphone or audio in device or run skype off a machine that already has a microphone built-in (like a MacBook).