Only 'Private' Available Defining Salesforce Opportunity or Case Access for a New Account Team Member

Salesforce Roles determine the access Users get to Cases and Opportunities regardless of who owns those records. The level of access is specified on the Role Edit page. Navigate to your User, click on the name of your Role, and select "Edit".. 

The list of initial possibilities when creating a new Account Team Member is determined by the User's Role setting.
If the Role's setting is restrictive (i.e. "Users in this role cannot access opportunities that they do not own that are associated with accounts that they do own"), only Private will show up.
If you want more options to appear you would need to edit the Role and go for one of the less restrictive options.

If, in the scenario above, you try to change the Opportunity level to "Read Only" you'll get an insufficient privileges error.

The only way to avoid the insufficient privileges message would be having Modify All Data or Modify All Records for the Object. Modify All Records permission effectively turns the User into a System Administrator with full access. Remember that you have multiple ways to control sharing (e.g. Organization Wide Defaults, Sharing Rules, Roles, Manual Sharing...) but that, once the access is granted by any of them, you cannot expect the restrictive ones to apply. In this example, having the Modify All Data permission will override the restrictions that are in place due to the Role settings or the Organization Wide Defaults.

A similar scenario is described in the following Article: Cannot add Account Team Members with read and write access to Cases or Opportunities