I have seen several reports of that issue in the last week on this forum and also one from a friend of mine.
Contact your cellular carrier customer support (and again and again if necessary). I recently helped a friend with the same problem. Her carrier is Xfinity (which uses Verizon's) network and the first two times she called she was told to contact Apple. The third time she called Xfinity she finally got someone who understood the problem and was able to clear her full voice mailbox. He told her that he cleared it by switching her account type to Android and then back to iPhone. He claimed the problem was caused by a recent Apple update that is notcommunicating with their system properly but I can't verify the accuracy of that statement.
The problem is not limited to Xfinity; I have seen recent reports from AT&T and Verizon customers as well.