Security issues making my head spin

I have the following scenario:

• Default BU = QA
• Additional BU = Social Services
• Team in Social Services = Medicaid

Role assigned to Medicaid Team:

1. Create - BU
2. Read - Parent/Child
3. Write - User
4. Delete - X
5. Append - BU
6. Append To - BU
7. Assign - User
8. Share - O

When I, as a System Administrator, try to assign a Case record to the Medicaid team, the system tells me

Insufficient Permissions. .. You do not have permissions to access these records.

What is that telling me???  And what is wrong with my configuration?  I am so confused by all of the security.