In some respects it does. There are certain fields and certain patient populations in which prestige or a "name" program (ie, a "name" to the lay public which does not necessarily translate into good training) means something special. This is why Dr. Ray on "Dr. 90210" constantly mentions his "Harvard training" despite the fact that he was only there 1 year. He knows people, especially the lay public and viewing audience will be impressed by that.
But beyond that, connections and relationships with colleagues are particularly important for those going into private practice (a dying breed, by the way). You will be somewhat reliant on the other physicians in the community to refer patients to you. Therefore, you might want to consider that it tends to be easier to set up a private practice in the same town in which you did your residency training in - you already know the attendings in town, the patient makeup, etc. So give some consideration to this when applying for residency.