Tweak the error message to make it clear that there may be > 1 duplicated value
Msg 2601, Level 14, State 1, Line 1
Cannot insert duplicate key row in object 'test.testErrorMessage' with unique index 'UXtestErrorMessage'. One duplicate key value was (Third).
Ideally you might have a bit more fill in the blanks and tell the number of violations if that information is available from the engine.