Case in point.
Harvard Med SAYS they want diversity. Having gone there, and still interviewing for them, etc., I can that this is not strictly true. By and large, the people who were there and from lesser known schools tended to be- my friend from U. of Missouri whose dad graduated from there and she was a nationally ranked runner. Several African-American students from Spelman or Morehouse who were the first person in their family to go to college, that type of thing.
It's harder to get into top schools coming from lesser known schools. That doesn't mean it's impossible or can't be done.