In explorer, when you open multiple files they all get a tab across the bottom of the window. Unfortunately the tabs are limited in size and shrink as you add more open documents.

I think for many of your clients, the file names probably start off very generic and have the key differentiation towards the end of the file name.  This key piece of the name is what gets cut off as the tabs shrink so you have to hover over each tab to see what file it is. It would be nice to right justify the text(name) in these tabs so it is easier and faster to pick the correct file. Another option might be to put the tabs at an angle so the full file name could be visible for each file.