I think the high school that a student applies from is a factor. It is not the largest factor in the admissions equation, but it is definitely a factor. Few colleges will come right out and say so, and there is no national data set I know of which can validate that statement objectively. I only know the data I see around me, which is the admissions results at one particularly well known public high school. Those admissions results are, frankly, not believable. So, there must be a flip-side. The only way that this particular high school can generate the admissions results that it does is if admissions are, in fact, biased to some degree by the prestige of the high school that the student applies from. That bias works in both directions–both for the student at some colleges and against the student at others. Those patterns are definitely visible in the Naviance data–colleges behaving outside of their published acceptance ranges year after year.
Yet another reason that it is important to be comfortable with your ‘likely admit’ schools.