This is interesting - I think I got it to work:
1) Under Settings > Notifications, Select Enable Email Alerts and Custom Email Alerts. Do not check Vibrate on Email or any other Email settings.
2) Under Settings > Advanced > Manage Rules, Setup your rules for email delivery and custom notifications.
For some reason these exact steps did not work last week. I should also note that today I had to recreate the custom rules because they were no longer working either. Very odd. Maybe there was a recent update.