This article will help when a user doesn't have permission to execute a Test. 

Problem

Check in the image above the associated error:

Cause 1

The User does not have the necessary issue permissions: "Browse" and "Resolve"

Solution

Go to Project Settings → Permissions → Project Permissions:


Cause 2

The Test Execution issue is not in a Status configured to disallow executions.

Solution

Go to Administration → Add-ons → Miscellaneous, under the "Workflow Status" section, to the option "Disallow executions of Tests with workflow statuses," and check if you have any status that does not allow users to execute the Test:

Cause 3

The Test Executions are restricted to the assignee user.

Solution

Go to Administration → Add-ons → Miscellaneous, on the "Test Run Options" section; the third one may be activated "Restrict tests execution to assignee," disable this option to allow all given users to be the Test Run assignee:

Check here for more information.