There's no need to join back to the table to do the DELETE. You can delete from the table by deleting from the CTE.

Here's the OPs original data in a readily consumable format...

--===== Create and populate a test table on-the-fly

SET DATEFORMAT DMY

;

SELECT [Timestamp] = CONVERT(DATE ,d.[Timestamp])

,Customer = CONVERT(CHAR(1) ,d.Customer)

,Data = CONVERT(VARCHAR(10),d.Data)

INTO #TestTable -- DROP TABLE #TestTable

FROM (

SELECT '3/03/2016','A','aaaaa' UNION ALL

SELECT '1/02/2016','A','bbbbbb' UNION ALL

SELECT '1/01/2016','A','cccccccc' UNION ALL

SELECT '1/12/2015','A','aaaaa' UNION ALL

SELECT '31/10/2015','B','bbbbbb' UNION ALL

SELECT '30/09/2015','B','cccccccc' UNION ALL

SELECT '30/08/2015','B','aaaaa' UNION ALL

SELECT '30/07/2015','B','bbbbbb' UNION ALL

SELECT '29/06/2015','B','cccccccc'

) d ([Timestamp],Customer,Data)

;

Here's the DELETE through the cte. Try it. It works just fine.

--===== Delete row in the table through the cte

WITH cteDelete AS

(

SELECT Customer

,RowNum = ROW_NUMBER() OVER (PARTITION BY Customer ORDER BY [TimeStamp] DESC)

FROM #TestTable

)

DELETE FROM cteDelete

WHERE RowNum > 3

;