I have had a situation where I would properly receive email notifications (both visual and sound) for all my emails in Thunderbird but one. That one was Gmail-based with a custom domain. Even though notifications for other two Gmail accounts I have would work properly, I would never receive them for that third one.
The reason behind this problem turned out to be in the Conditional Store (constore) facility which provides a protected update mechanism for message state information that can detect and resolve conflicts between multiple writing mail clients. It also allows a client to quickly resynchronize mailbox flag changes.
Thunderbird still doesn't have a full support for that feature and you will have to disable it. Go to Menu > Options > Advanced > Config Editor and search for "condstore". Double click on "mail.server.default.use_condstore" to set it to false and you are done.
Now just restart the Thunderbird and send yourself a test email to make sure notifications are working as they should.
