One thing to keep in mind is that these "tiers" COMPLETELY depend on what field you're in. For my field, Princeton, Columbia, Berkeley, Penn, and NYU are all definitely 2nd Tier schools.
I would definitely not consider all the Ivies to be "top tier." This is not college, so having an Ivy League name does not guarantee anything. As you said, Cornell in particular does not have a great reputation.
Finally, consider how happy you would be at these places. Columbia & NYU have the reputation for being very competitive and having unavailable professors. I almost went to Columbia, but decided I would be very unhappy there. Think about where you would do your best work, because ultimately what matters is not whether your school was ranked #5 or #7, but how good your scholarship is, how much your advisor can do to help you while you study under him/her, and how much s/he can help you make connections and find a job.