Hm, that’s odd.
In this case, I recommend upgrading manually: https://codex.ww.wp.xz.cn/Upgrading_WordPress_Extended
It’s pretty straight-forward, the guide is very detailed, and it will hopefully resolve the issue for future updates.
It’s safe to go straight to 4.0.1 from 3.9.3.
Make sure you aren’t running any weird server caching tools like Memcached or APC. They get ‘stuck’ sometimes.
I think it may have to do with my permissions being messed up on this server. I am not able to load plug in in the normal way, I have to upload them.
But in the past, the core updater has always worked.
With the permissions, make sure that all files are set to 644 and all directories are set to 755.
You can use your FTP client, the instructions vary for each, so I can’t really offer specifics.
https://codex.ww.wp.xz.cn/Changing_File_Permissions
Thanks – I’ll try this over the Holiday.