What have the results of these negotiations been? What is the median % increase over the initial offer?
Some of the companies you mention try hard to make the initial offer compelling/fair (e.g. slightly above max of {current compensation, competing offer, target compensation for level}), so I'd assume that sometimes they simply won't budge beyond that, as they feel that they've taken into account all relevant info before making the offer.
I look forward to hearing from you soon!