I found the answer here:
- If the install uses FTP for updates (and prompts for credentials), automatic updates are disabled. (I found this answerthis answer which will tell you how to avoid using FTP.)
- If the install is running as a SVN or GIT checkout, automatic updates are disabled
- If the constants DISALLOW_FILE_MODS or AUTOMATIC_UPDATER_DISABLED are defined, automatic updates are disabled
- If the constant WP_AUTO_UPDATE_CORE is defined as false, automatic updates are disabled
- Your WordPress install also needs to be able to contact WordPress.org over HTTPS connections, so your PHP install also needs OpenSSL installed and working
- Wp-Cron needs to be operational, if for some reason cron fails to work for your install, Automatic Updates will also be unavailable
This plugin will test these things for you.