I've noticed I occasionally get an e-mail message about a direct message being sent to me over BitBucket, but when I look at my BitBucket inbox (link to it provided in the received e-mail message), the message is not there.
My guess is that if I delete a message from my BitBucket inbox and then someone sends me a reply to that message, their reply will be sent to me as an e-mail message but will not be added to my BitBucket inbox.
This is, of course, wrong.
E-mail is not a reliable messaging medium, as the message could in theory never arrive and without you/BitBucket being notified about it. On the other hand, BitBucket inbox and on-demand web access to it should be reliable and should not allow messages to be lost without the owner/me explicitly deleting them. I could perhaps not be able to access the inbox at any specific point in time, but once I do access it, there should be no way a message is missing from it.
Hope this helps.
Best regards, Jurko Gospodnetić