When I delete voicemail messages, they reappear. My inbox is constantly full?

Don’t blame Apple for this; it’s a carrier problem. Either go online to access your voicemail inbox and clear them there, or contact your carrier for assistance (this happened to one of our users on a company phone).

Hey cmarholz!

I have an article for you that will help you troubleshoot this issue with your voicemails:

iPhone: Troubleshooting voicemail

http://support.apple.com/kb/ts4430

Take care, and thanks for visiting the Apple Support Communities.

-Braden

That doesn't work..... iOS7 has ruined a perfectly decent voicemail system.... After deleting the messages one-by-one, instead of marking bubbles and doing them all at once..... it refills.

I've wasted more time, with iOS7 mail services.... Such a "pretty" waste of time....

I wonder if the software engineers at apple ever read these posts. Would love to vent with them.... iOS7 is pretty, but it has "blown up" my iphone5.

I'm on the verge of becoming an iphone hater....

Hey Braden!

I have an article for you that won't help anyone troubleshoot this issue since, as you can see, it doesn't cover the issue raised in the OP:

iPhone: Troubleshooting voicemail

http://support.apple.com/kb/ts4430

Take care, and thanks for visiting the Apple Support Communities.

-shmeck

Put your phone into Airplane Mode (Settings > Airplane Mode switch to ON (green side showing), then delete the messages. Switch Airplane mode back off. You can also call in to your voice mail service and delete them from there.

I’m having the problem now, in 2017 on my brand new 8plus. The airplane mode does not help the problem!

people can’t leave me voicemails now because it’s full!

Apple you are ruining your products!