Is there a bias in top tier PhD admissions at top tier universities to favor those with undergraduate degrees from other top tier universities?

Out of 9 applications to graduate schools ranked in the top 25 according to US News, I was accepted to only one PhD program in Electrical Engineering, and just barely (see note below). According to my GPA, GRE scores, letters of recommendation, and previous relevant research/industry experience I thought that I had a much better chance of being accepted to these programs. The only thing that I can see that sets me apart from many of my peers is that my undergraduate alma mater is pretty obscure (barely top 200 R2 university).

Here's some extra information about my application in case something else here sticks out:

Undergraduate Majors - Dual Computer Science & Electrical Engineering

Undergraduate GPA - 3.97

Research/Industry Experience - Two summer internships at NASA resulting in two conference publications. Four years as a professional Electrical Engineer doing R&D for a defense contractor which resulted in two more conference publications.

GRE - 166 Quantitative, 159 Verbal, 5.0 Writing

Maybe these schools are just monumentally difficult to be accepted into? I'm curious if anyone here has some insider insights.

Note: The reason I say just barely is because, after not hearing back from most schools, and being rejected outright from 3, I called the one I was most interested in going to and told them if they still had me on their list of viable candidates and were willing to make me an offer that I would accept. I received an offer letter the next day, two weeks before the deadline. I received rejections from the remaining schools not long thereafter.

Answer

Excuse my short answer: absolutely.

Obviously, a school with a good, or even a great name, will give a candidate a big boost in his/her application. This is a rational decision: the recruiting committee wants to minimize any risks and they don't have much time to delve seriously into the history of each candidate. So if you graduated from a top school with excellent credentials you will have an edge over those with equal credentials but lesser perceived schools.

Good references from colleagues of faculty in the department you're applying to, and good ties with your school are also strong factors for admission.