Though I'm applying to graduate school for next year, I still like reading this stuff
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Anyway, having gone to undergrad at an Ivy, I would say that the whole cut-throat aspect is reserved more for the undergrad experience (where kids, having had it reinforced their whole lives, value themselves only on their successes and good grades), and even then only within certain majors or pre-professional paths (think Business, Pre-Med). When I was working in a lab, I never sensed a spirit of competitiveness between graduate students. They were all focused on themselves, on their projects, going at it alone. It seemed like a lot of PIs were juggling too many roles and hats - writing grants, being on committees, consulting on the side, having to go to DC to be on some President's committee, etc. In other words, not a lot of the PIs seemed to be consistently there, in the lab, if a graduate student wanted more robust mentorship experiences. On top of that bureaucracy was a nightmare, and a lot of the science programs are housed in old, unrenovated buildings; just because the school has a large endowment and brings in NIH research dollars doesn't translate to good stewardship of those resources.
Personally, I'm going to apply to a few select Ivys only because of their programs and the faculty in those programs, not because of a preconceived notion of competitiveness at the graduate level. The interview will be important for fit. However, I will, for the majority of my applications, be applying to top non-Ivy schools just because I would also like a campus that might be larger, friendlier, happier, and more likely to be investing in new facilities (to attract high-caliber faculty and students).