Logs Table - Assign Serial Number To Techs

I have Logs table and want to assign a serial number to the techs using the following query

Select
Date,
Case_ID,
Site,
Dept,
Tech,
Start_Time,
ROW_NUMBER () OVER (PARTITION BY Date, Site, Dept, Tech ORDER BY Start_Time) as Row_Num
FROM
Logs
Where Date = Getdate()

I get the following results.

Date Case ID Site DeptTechStart TimeRow_Num
7/28/14 10023 TartvilleMaintcAmy P.7:301
7/28/14 56789 TartvilleMaintcRem W.8:051
7/28/14 23098 TartvilleMaintcAmy P.8:352
7/28/14 70004 TartvilleMaintcAmy P.9:103
7/28/14 12708 TartvilleMaintcMag O.10:001
7/28/14 10004 TartvilleMaintcAmy P.12:304
7/28/14 40056 TartvilleServiceJoe F.7:301
7/28/14 23458 TartvilleServiceJoe F.7:552
7/28/14 69200 TartvilleServiceRus T. 7:301

Please notice the cases in Maintc department. See how Amy P.'s shift is broken by Rem W. and Mag O. But the Row Number does not recognize this, it still says Amy P's case as 2 and 4 the even though Rem's and Mag's cases were in between.

This is what I really wanted.

Date Case ID Site DeptTechStart TimeRow_Num
7/28/14 10023 TartvilleMaintcAmy P.7:301
7/28/14 56789 TartvilleMaintcRem W.8:051
7/28/14 23098 TartvilleMaintcAmy P.8:351
7/28/14 70004 TartvilleMaintcAmy P.9:102
7/28/14 12708 TartvilleMaintcMag O.10:001
7/28/14 10004 TartvilleMaintcAmy P.12:301
7/28/14 40056 TartvilleServiceJoe F.7:301
7/28/14 23458 TartvilleServiceJoe F.7:552
7/28/14 69200 TartvilleServiceRus T. 7:301

I tried many combination of columns for Partition by () and Order by () and the best I can get is the result at the top. How should I achieve it.