That shouldn't be the case if the server has a big enough TCP sender buffer/window.
Mat, the one really common case where multi-part transfers make a huge performance difference is when your ISP use "shaping" of your traffic. This is becoming increasingly common. In practice, it means that they will limit your speed per TCP session to something much lower than the total bandwidth you are paying for, so that you have to have many TCP sessions going at the same time to reach the full potential of the line.
My ISP does this on my 100/100 fiber optics connection to my house in Sweden, and my girlfriend (who lives in the US) has the exact same problem with her ISP. We both use the multi-part feature extensively to transfer files between us.