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So I negotiated somewhat successfully with a T20, I guess, but the extra 4k/yr(to match another T20 offer) doesn't seem to make that big of a difference in terms of total coa.

Additional 20k, for instance, would be nice. But I don't know if asking again makes me look like an ungrateful person that they no longer want. Any thoughts? Has anyone negotiated for some more successfully for the second time around?

What confuses me more is that the extra scholly not part of the merit award, it's named after some person (I don't know if this fits the criteria of a "named scholarship"). Is this normally how it works for people who negotiate scholarships? Or is this something special?

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I was literally just thinking about this same thing. Really people have re-negotiated without anymore ammo? It does seem like a weird thing to me, but I guess a school would rather have the option to give you more or not before you flat out withdraw?
Should you send an email like "Thank you *nice stuff nice stuff nice stuff* but if my offer is not raised I will, 100%, choose another school. This is not a hardball tactic, simply the truth that I will not accept the current offer, *nice stuff nice stuff nice stuff*"?