Ok, I'm going to try to shed some light using two sources I picked up applying in the last cycle. Columbia gives a student handbook to interviewees that lists the undergrad institutions of each class and Cornell gives a list of accepted students to accepted students.
Here's where a few students chosen randomly from Columbia's class of 2007 went for undergrad:
Notre Dame, JHU, Yale, Messiah, Princeton, Temple, Columbia, BU, Georgetown, Columbia, Harvard, Union, Princeton, Columbia, Princeton, Harvard, Yale, Morgan State, Harvard, Michigan, Oberlin, Virginia Polytech, Stanford, Fordham, U Wisconsin, Georgia Inst. Tech., Dartmouth, Case Western, UC Berkley, BYU, Yale...and I'm tired of typing
Here are a few from Cornell's accepted class of 2008:
Columbia, Cornell, Univeristy of Florida, Columbia, Penn, Oberlin, University of Virginia, Cornell, University of Virginia, Penn, Stanford, UC Berkley, Princeton, Princeton, Cornell, Florida Agricultural and Mechanical University, Bowdoin, Yale, Harvard, Harvard, Harvard, Davidson, USC, Cornell, Univeristy of Florida, Harvard, Cornell, Columbia, Stanford, Harvard, University of Wisconsin-Madison, JHU, Middlebury college, Tulane, Stanford, Santa Clara University...and I'm tired of typing again.
Now I'm wondering why I typed all that. Looking over the list it seems like 20-25% of the class at these schools are not from big name undergrad institutions. Of course "big name" is comepletely subjective. Personally, I think most of the people who get into the most competitive schools are people who differentiate themselves from the masses of applicants in some meaningful way. I would expect this to be especially true of applicants from less-well-known schools.
