Here is the reply from Zimbra support:
Jason, we don't test the Zimbra Connector for Outlook (ZCO) with other outlook plugins installed, so there's a possibility that the BB connector to outlook and the ZCO are conflicting with each other.
We do offer a solution for OTA sync with a blackberry, but it requires another component (analogous to the BES) which may not be feasible to support one user.
I would say that wired sync from Blackberry Desktop Manager 4.2 to Microsoft Outlook 2003 does not work. Does this mean Zimbra is storing the data in their own files and not letting MS Outlook manage its data?