Custom Records is not Available in the Import Type field of the Import Assistant

To resolve this, perform the following steps (Administrator role is required):

     1. Navigate to Setup > Users/Roles > Manage Roles.
     
     2. Edit the custom role used for importing.
     
     3. Click the Permissions tab.
     
     4. In the Lists subtab, add Custom Record Entries with Level set to Create (or Higher)
     
     5. In the Setup subtab, add Import CSV File with Level set to Full.
     
     6. Click Save.
     
     
Note: Individual permissions added under the Custom Record tab does not give the user access to import custom records of only that type.