Yes, they do apply for additional financial aid and also for scholarships that are outside the schools, like some foundations or institutions, who give aid for people who study certain topics. When you apply you have to check if there is additional essay for people who want to get a scholarship, or an assistantship and add this to your application. They will also inform you if the need such assitional materials while applying to the chosen schools. Also, mark on your choice lists the reasons why you want to apply and the scholarships you want to apply for (TA, or other financial aid) so they will know what you want.I have to admit, I aimed at schools that I know I could afford within the limit of Fulbright grant - so that was bit on the safe side. I just dont have any additional dollar to spare. But you certainly have to dream big, because good schools have good financial aid :-) Maybe, knowing what I know now, I would add a famous school to my list, but only one