'Insufficient Privileges' Salesforce Error After Clicking 'Authorize Connection'

If you click the 'Authorize Connection' button for the Salesforce integration...

...and receive the following error in your Salesforce instance:

Insufficient Privileges
You do not have the level of access necessary to perform the operation you requested. Please contact the owner of the record or your administrator if access is necessary. For more information, see Insufficient Privileges Errors.

...then please ensure that you are not logged into your Salesforce user such that a banner at the top of the screen reads "Logged in as..."

Please logout and login directly as the desired user.