A user’s security role needs to match the privileges at the ‘User’ level with the Access Team Template Privileges (R,W,A,A2,D,Assign and Share) associated with the Entity. If an Access Team has Delete privileges and a user's security roles did not, the user will be blocked from being added to the team. CRM Security is AWESOME and this shows that security can’t be bypassed just by adding someone to a team. If that user cannot delete a record, then adding them to an Access Team with that privilege won’t work!
Big thanks to the awesome: Thymio Barbatsis, for continuing to share incredible tidbits with the community