Recently I had this happen twice without really knowing why. The add new contact button was missing on the case form. In the old UI it was present, but in the new UI it was gone.
Problem
Luckily support found out that this is caused by the contact field containing some old “metadata”. All you have to do is remove it and add it back.
FIX
As you see here we have added the same field twice.. Now you can remove the “OLD” contact field and then publish..
Voila!
A pretty random thing to happen, but an easy fix:)