In order to be able to modify Test case results, e.g., Status , Comments, using the Test case editor or Result grid view, you must have "Read/Write" access to the the Test or Test Suite item containing the Test Case(s). It is not sufficient to have Read/Write access to the Test Case(s) themselves. This is because the results are saved in the context of the Test. 

Example Data

To check, select the Test or Test Suite, 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 or Test Suite item is assigned. The user will need a minimum of "Items Write" access to the library.