We've had Century Link voicemail for 10+ years. Today it stopped working. I'm putting up this info for any others who have this problem.
Calling CL repairs didn't help. I tried three times. The first two reps disconnected me. The third figured out it was because voicemail had been removed from my account. He transferred me to billing.
Billing said somehow (?) call forwarding had been put on my account. That supposedly disabled voice mail. The billing person had to remove that feature and add voicemail back on. He said it would take until Monday for this "new feature" to be effective.
At first, this increased my monthly bill by $10/month. The billing guy spotted this and corrected it. If you go this route, double check yourself any statement of charges they show you for confirmation.
Good luck.