I don't think the top 20% is the appropriate demographic to study in gauging how much tax the richest Americans pay relative to others. The last figure I saw that I know was legit was from the '90s, when it was reported that 10% of Americans accounted for 90% of the wealth. I would venture to guess that gap has only widened since then, but I don't know the current numbers, so I'll stick with those.
If the top 10% account for 90% of the wealth, the other 10% of those being placed under the 20% "rich" umbrella are surely accounting for a much smaller percentage of wealth than the top 10%, yet they are being taxed at the same rate as the top 10%. If I were one of those people, I would be really upset with this tax structure. Being among the 30% that aren't called "rich" and still pay some amount of income tax, even I don't like seeing that 50% aren't paying any tax at all, but I think that top 10% is definitely getting over on the rest of us, especially the 10% immediately below them.
I think that second tier (top 11-20%) represents the folks who will tell you they worked hard for what they have and shouldn't be shouldering the burden of those who aren't contributing. For those folks, that argument makes sense to me. On the other hand, I'm pretty sure there are very few in the top 10% who have amassed their own wealth through hard work and dedication. Most of them were likely born into rich families and have never lifted a finger, except to make investments that earn them more money. To look at this from an honest perspective, I think we should only qualify the top 10% or so as "rich."