Tom’s answer looks really useful to me.
I’d also suggest the standard WP troubleshooting tip of turning off plugins and seeing if you are able to add categories when your plugins are off. If so, turn plugins back on one by one and see if you can figure out which plugin is the “culprit”.
Sometimes it’s a plugin that is intentionally doing something to manage your categories for you, for instance, and sometimes it’s a plugin that’s just causing a glitch. Often you’ll be able to get help from the developer if it’s a true conflict and not a feature.
Best of luck!