You: "So the Brownsville White People huh?"
Local: "Yeah man, they've won 3 national championships in 5 years! We love the White People! Hey, you're white, don't you feel honoured? Kinda funny right, considering white people can't jump! Just promise you won't break out any dance moves when we win a 4th title! Hahahahha! Oh, I'm only joking".
Of course, something like the name of a sports team is only a small, small part of institutional racism. Now, imagine that in this place, certain Hollywood westerns have become quite popular. Now every time there's a white person in a local movie, that character will always just happen to wear a cowboy hat and be good with guns. Heck, the new Brownsville blockbuster hit has a white female lead! The Brownsvillian main character falls in love with a modern Texan girl, represented as a cowgirl. This girl's family are myopic and violent. But of course, the main character manages to beat up all the white people and get the girl in the end. What a salute to your culture! You can't even walk into a bar without the bartender doing making a little pistol gesture with his hand, greeting you with a pretend gunshot. As fate would have, you're not from the south, heck you're not even from the States, you're from a small, middle-class family in London. You've never even set foot on American soil before, so of course you don't really feel any connection from seeing a cowboy on film. Not that you'll ever try to explain that to the bartender though -- he was just trying to be friendly.
Imagine all this and understand that this is reality in which minorities have to live in within western countries. So no, the native Americans aren't just getting butthurt over nothing. This Redskins thing is just one small bit, something small that can actually be changed, that's part of the bigger, institutionally racist picture