It really isn't enough to just submit your resume and wait. Big companies receive thousands of applications and it is all too easy for yours to get lost in the shuffle. First make sure your resume is in tip top form. Maybe have it rewritten by a resume writer. Then maybe try to talk to the hiring person directly and try to schedule an interview.

There is nothing wrong with starting at a smaller company and then working your way up to the big boys. While you are working with a smaller company try to network with people who work at the company(ies) you want to work for.