There is a "Company Domain" setting in the Case Flags Setup that specifies that a Case flag will be set if the email domain does not equal the company domain specified. This means that if an internal user with the same company domain responds to the case email, a case flag will not be set. 

Example: A company domain of vicasso.com is set in the Company Domain field in the Company Domain setting of the Case Flags Setup. Jane (jane@vicasso.com) sends a message regarding case 001. Bob (bob@vicasso.com) responds to the case email, but no flag is set on the case.

The solution for this scenario is to clear the Company Domain field in the Case Flags Setup. If there is no Company Domain listed, internal users will be able to set the Case Flag by adding a comment to the case.