Hi Rutherford,
To be honest, I'm not all that familiar with schools' "back door dealings" type of admission activities. (I wasn't a well-connected applicant when I applied and most of my clients aren't either.) However, if one of the trustees, overseers, or board members of a particular school actually knows you well (and hasn't just met you a few times and had polite small talk over cocktails) and can vouch for your candidacy, you will want to consider having them write one of your letters for you or perhaps even making a call to the admissions office.
Your question will probably be answered a bit more satisfactorily by someone with more experience with these kinds of invisible modes of influence—but I also doubt someone with that experience will post on a public forum.
My short answer is: Go ahead and use the connection if s/he actually knows you well. Otherwise, it's a waste of time and could potentially make you look bad.
Best of luck with your application!
Kaneisha
Kaneisha Grayson, Harvard MBA + MPA '10
Founder, The Art of Applying®: get into the graduate school of your dreams and get money to pay for it
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