Mail notifications not working on Mac despite enabling settings

Why isn't my mail notifications coming through?

I made sure to turn on Badges and Alerts on in System Preferences and ticked all the other boxes.

Also checked Mail's Preferences and ticked or selected an option for the following:

• Check for new messages: Every minute
• New messages sound: New messages sound
• Play sounds for other mail actions
• New Message notifications: Inbox only

And still after checking my inbox (both personal and school) after a couple minutes, still no notification, checked the my mail and I had a bunch unread that I wasn't notified.

Please help and Thanks!

[Re-Titled by Moderator]

Original Title: Mail notifications not working?