Well, it turns out there were really 17 voicemails on my phone. Not exactly on my phone, but on the phone that is supposed to forward to my phone. Which means that the people calling were spared my long "do not leave me a message" message but also did not hear those instructions at all and left a lot of messages.
Evidently, sometime after the first of the year, my phone got un-forwarded. Had it not taken more than 2 weeks to figure out how to forward the first time, I would not have been worried. It seems to have been resolved now, but I still have another 16 calls to return!