It seems that the inbox on MSO still isn't refreshing properly. I believe the initial pains are supposed to be fixed, but it's difficult to see in the huge feedback thread, so I'm reporting this separately.
I had a tab open on MSO and another tab open on another SE site. Someone commented on one of my posts, and I saw a notification appear (red ①) in the inbox on both sites. I decided to open the MSO inbox, probably less than a minute after the notification appeared. When I clicked, the list of replies did not include the one that triggered the notification: the latest reply in the inbox as it appeared on MSO was my then next-to-last reply dating from a couple of hours ago. When I opened the inbox on the other SE site, I saw my latest reply as I should have. After refreshing the MSO tab, the inbox showed my latest reply.
This doesn't happen all the time: I've also observed the inbox to refresh properly. So far, my observations are consistent with the following:
With the new top bar, if I've already viewed my inbox in a tab, then when a new notification comes in, the red ① is displayed as expected, but the list of messages displayed by the inbox is not updated.
Google Chrome 30.0.1599.114 if it matters.