I have noticed in the web client that if you have a message with an attachment to it that is in a folder that has been shared to you, you do not get the option to save the attachment directly to the briefcase.
For an attachment in your own mailbox you get links for "Download | Briefcase | Remove". If the attachment is in shared folder you just get "Download | Remove".
I am having a difficult time determining if there is any good reason for why this feature would be missing. Given that you have permission to retrieve or delete the attachment, it doesn't make much sense that you shouldn't also be able to directly deposit the attachment in the Briefcase.
I have noticed this in installations running 6.0.8 and 6.0.9.