Aawwwwyeahhhh!! I grew up a just east of BB and a little less than a mile off the border, needless to say, there wasn't much land that I didn't cover between horse or hilux. @linuxgod if you want, I can put together a pretty remote trail route for yeah. It's going to be something that will definitely put you in places that I'm sure very few people have explored, but really will be an experience as you will feel you're in the 1800s. Just make sure you have a sat phone if you go.
Now I know a lot of people all say things like, "well it's dangerous running the border" and stuff like that, it's not, just need to be a little smart about if your running into illegals, or if you're running into drug mules. Illegals crossing as no big deal, it's possibly interfering with drugs that will get you messed up. But I do sincerely recommend bringing a firearm if you want to roam past Big Bend, preferably an AR-15 (or any other common semi-auto, I mean, they all function the same, so take your pick).
For others, I know someone might say I'm being paranoid, but seriously, most here know that I have a story for everything I do. This is no different, it's beautiful big sky country, but it ain't called the bad lands for nothin'.