I solved this problem on my phone by completely erasing the device and setting up as a new phone. I had to reinstall iOS 10.0.2 (although now you would probably have to reinstall the updated 10.1).
1. I saved all my contacts and photos through iCloud backup.
2. Erased the phone connected through iTunes on my mac
3. I set it up as a NEW phone.
4. Reinstall and set up everything from scratch [big time pain in the neck].
I initially tried it the other way, "Restore from backup", but that didn't work--kept getting the voicemail glitch that way. Setting up as a new phone did work.
It was kind of a pain, because then I had to reinstall all of my apps and ring tones individually. I also had to set up all of my email accounts, etc., all over again. It was annoying, but I haven't had any issues since doing it that way. Hope this helps.