I currently have an Exchange 2013 mailbox that is shared by 15 staff members. I set it up to work in online mode due to the constant flow of e-mails, and for the most part it works fine (Mailbox is about 40 gigs right now). All these staff have their personal mailbox in addition to the shared one and run Office 2013 or 2016.
However, these last three staff members I added in the past few weeks have been having similar issues. Mail will appear to get stuck in the Outbox and not get sent. It also seems like they're disconnected on occasion from the shared mailbox. Closing and re-opening Outlook sometimes helps, but it's an ongoing issue.
I tried recreating the user profile but that didn't help. I don't know if there's particular settings I should look into tweaking in Exchange? The staff members that have had this mailbox for awhile don't seem to be affected which is strange as well.