Error Code [28:341] Can not connect Exchange Server for one of these reasons

Connection to Exchange Server fails because:

One or more Exchange services are down.
The Service Account does not have sufficient rights.
The Static profile does not exist.
Auto-discover config issue.

Symptoms

Exchange mailbox backup job fails with [28:341] error code.

Exchange mailbox backup job fails to log on to the server for certain mailboxes.

Causes

Various causes relating to the Exchange mailbox account used for agent access or to connection issues from Outlook on the agent host computer.

Resolution

Verify the following:

The user account set to run the Exchange Mailbox Agent should NOT be a member of any other groups except these three:

Domain Users (AD)
Organization Management group (AD)
Local Administrator (Local machine where the agent is installed)

WHY? Because any other groups assigned to this account could cause this account to inherit DENY on "Send-As, Receive-As" which is required for this type of mailbox access. User ADSI edit to confirm permissions. See Documentation's Prerequisites - Exchange Mailbox Agent

Make sure the Outlook version running on the agent host computer has the latest service pack and hot-fixes to avoid MAPI issues.
Make sure Outlook remains closed on the agent host computer during backups.
Check Outlook to make sure there are NO antivirus add-ins loaded. If any, please unload them (disable loading of AV outlook add-ins). You really don't need AV scanning on backup.
Check that Outlook is not suffering intermittent disconnects. Refer to the Windows Application Event logs on the agent host computer and search for keywords related to Outlook events.
Check that the Outlook profile used is not in Cached mode. In some cases, recreating the profile may resolve the error.
Check that CAS Server Name in the agent configuration is set to fully qualified domain name and not the short hostname.
If the backup fails at a specific mailbox account, check access/validity of the account. If invalid, exclude or delete the account.