In order to be able to modify Test case results using the Test case editor view, you must have "Read/Write" access to the the Test containing the Test Case.

Example Data

Although the Test Case has a Status of "Work", if you are unable to edit Test case results, e.g., Status , Comments, it is likely because you do not have the proper access rights. 

To check, select the Test and check the Access in the Overview view. 

Test

Test Case

In the above example, the user only has "Read Only" rights to the Test. This will result in the user not being able to modify the Test case results even though the user has Read/Write access to the Test Case itself. To modify the access to "Read/Write" contact someone with "Change" rights to the library where the Test item is assigned.