Hi!
I have worked for several years for both BCG and McKinsey and covered different geographies at both firms. I honestly believe that this is nothing but a cliché if you relate it to the interpersonal behavior of the people. There is really no way you coan claim that either of the two are behaving more/less elitist/snobby/whatever at the aggregate level. Spot experience might of course vary a lot, depending on who you ask and his/her subjective experience.
That being said, there are a couple of deeper, "institutionalized" patterns that most people are not aware of, from which you could infer that McKinsey is less concerned with looking at what BCG does than vice versa. I can see this in alumni meetings that I attend - BCG folks are always very eager to hear from me how McKinsey was and whether/how it was different. At McKinsey meetings however, it is never really a topic of interest that I also worked at BCG. Another hint that speaks to the same trend is the importance of the "Why BCG/McKinsey?" question in interviews. For BCG, this is really important and a candidate should have a very strong and convincing answer why he/she prefers BCG over anyone else. At McKinsey however, this is not one of the "make or break" questions (unless you say something really stupid obviously) - you could interpret it as McKinsey sort of assuming that of course they are the candidates' first choice and don't need him/her to proove it (which of course could be called "elitist" ;-)).
Hope that helps, Sidi
(edited)