Graduate School Catch-22

I’m in the process of applying for graduate school in the biological sciences. I’m Canadian, and I’m applying to both American and Canadian schools. I’m considered to be an international applicant to the US schools, and therefore I’m required to fill out an official statement of finances with my application.
The statement carries a warning in big bold print :

Please note : your admission application will not be processed without this form and the required certification letters.

Basically, it demands proof that there are 14,000$ US sitting in someone’s bank account and that this money is for my school.

Now, I’m your average penniless student, and I’m applying only to schools that provide a stipend to graduate students, or teaching assistantships which would cover tuition and living expenses. There is a lovely example of a catch-22 here and I’m not entirely sure how to get around it.

You need to get an assistantship/stipend from the department to pay for school.
You need to be accepted in the program to get an assistantship/stipend.
You need the assistantship/stipend (and proof of $$ available) for them to initially consider your application.

I’ve e-mailed some of the schools involved and I’m still waiting for answers. I found calling to be useless, because of endless press-1, press-2, leave-a-message games.

If anyone on the SDMB has gone through something similar and would like to share secrets of how to get into these well-defended American grad schools, I would really appreciate the help!