Salesforce Data Import Wizard Error 'You do not have sufficient permissions to import custom objects'

When using the Data Import Wizard to create records, the error "You do not have sufficient permissions to import custom objects" may occur due to permissions but also in the following scenarios:

• Import custom objects via a CSV file.
• Import custom objects without creating a custom object in the organization first.

Steps to reproduce:

1) Setup | Data Management | Data Import Wizard
2) Click Launch Wizard!
3) Under the 'What kind of data are you importing?', select Custom Objects
4) You see the error message described.