My application experience has proven that a school's brand name matters quite a bit. I went to a lowly ranked undergrad/MBA before applying to PhD. My resume would show that I'm a hard worker, I have very well written SOP's, fairly well thought out research ideas, glowing recommendations, and competitive test scores. I was not offered a single interview, much less admit, by any top 20 PhD program. After speaking with some of the faculty at these programs, it seemed like I passed all the initial screening rounds on paper, but none of the schools were willing to take a chance on me because of the signal that a poorly ranked undergraduate institution sends.
I was accepted by every school that offered an interview. At each of those interviews, the interviewer's first matter of concern was my academic pedigree. These questions usually started with, "you have great GMAT's, why did you attend xxxx?" Directly related to this concern were the concerns about the rigor of my academic preparation. In short, if you went to a school with a good brand name, you won't be subjected to these doubts during the admissions decision that can lead to cognitive biases, i.e. halo effect.