This is no longer an issue with the new interface, go SE devs!
When I click on the Stack Exchange menu to get at my Inbox and click on one of the links, it will open on a new tab which is great. However, the menu itself will remain open in the original tab which means that I won't receive notifications of new messages until I remember to return to that tab and manually close the menu. As long as I have the menu open in any of my browser's tabs, new messages/notifications are automatically marked as read.
This has led me to twice (I deleted the second when I realized I had left the menu open) post meta questions over on SU to report what I thought was a bug in the notification process. I agree this is a case where the user (me) was not being particularly intelligent, but it seems reasonable to have the menu close when the tab in question is not active. That way stupid stupid users like myself won't think there are bugs where none exist.
EDIT: @Doorknob correctly highlighted another issue. At the moment, any new messages that appear when the menu is clicked will be marked as read if the menu closes irrespective of whether they were actually read. This means that if my suggestion is implemented and the menu closes when it loses focus, it will be hard to distinguish read from unread. So, my suggestion is:
Make sure the menu never remains open in unfocused, backgrounded tabs. This ensures that I will always have a visual hint when I receive new messages:
Only mark new messages as read if that actual message has been clicked on. That way, if the menu closes in the background, I will still see unread messages in bold and have a visual indication of whether I have read a message or not. In other words, don't mark messages as read when the menu closes but only when they have actually been clicked on.