I have seen this happen before. Sometimes it’s been related to the web server. Host occasionally have trouble that prevents new files from uploading properly. If your host has a status board where you can check on their system, it’s worth a look before pulling your hair out.
If you don’t find anything wrong there, then I would check with a FTP program to see if the files are being uploaded to the server correctly, even if they don’t show up in the media library. WordPress will typically give an error if there’s a problem here, but occasionally that won’t happen. You’ll want to check the wp-content/uploads folder and see if it’s adding the files there.
After that, I would try deactivating all plugins and see if you can successfully upload then. If you can then turn them on, one by one until you can no longer upload files. Then you’ll know the source of the trouble.
If the plugins don’t do it, try switching to one of the default themes and then give it a shot. If it still won’t go, the last thing I would try is going to the updates section in your dashboard and re-installing the current version of WordPress. That could help if something has some damaged one of the core files. Test it out after that.
Should you still not find the trouble then, you’ll probably need to ask your web host. You will have ruled out plugins the theme and have a new start with the WordPress core too. Hopefully they’ll be able to help you out.
Let me know what you turn up and if I can help,
Bill