Very late answer but it might help someone else trying to achieve this.

You can do this by editing the html of the application page (click on 'Customize -> HTML/CSS Editor') and duplicating the div element that relates to the Resume attachment, see image below.  You just need to change the references to "Resume" to something else in the duplicated section ie. 

for="application_letter">Letter of Application<

and

name="application_letter"

Doing this will create a second upload field, which will in turn create a second attachment in the application record with the name 'application_letter'.