No, you cannot.
Email groups – also known as <b>Distribution Groups</b>, are not email accounts. Mail sent to Distribution Groups is sent to all members of that group. Distribution Groups do not have a mailbox of their own.
What about security groups? You can most certainly apply a mailbox to them and the e-mail will go to each member.
It would work if you set up mulitple email boxes for the single user and had the primary MB forward a copy of the email to the other. Then sync one CB to one MB and another to the other. Not sure what to do about contacts or calendar. Maybe share them between the two MBs.