20 mile days are definitely possible. 2 weeks of 20 mile days back to back is a bit harder. The only things a person really needs to do this, though, are a really high tolerance for pain and an ability to fend off boredom. It's not as easy as it seems--not nearly--but a mantra that helps me sometimes is repeating to myself "if I can do this for an hour, I can do it for another hour." In all honestly, I do believe that the majority of the hikers who attempt a thru on the AT are deterred by boredom (or the prospect of several months of boredom).
Hiking 20 miles in a day really isn't that hard. All you've got to do is wake up at dawn, get on the trail as soon as you break camp, hike all day long, and make camp at dusk. In the Spring, when days are about 12 hours long, I've done 25 mile days at a pretty leisurely pace. It's the "hike all day long" part of that simple description which can't be underestimated. A relatively light pack is a given, and it's assumed that you're healthy enough for backpacking if you're even considering 20 mile days. The AT certainly has some elevation gains, but take these slow enough that you aren't sweating very much more than you would on flat terrain and you'll be just fine with your schedule.
So let's assume that you can do this without tiring yourself so much that you have to take a rest day in the course of the 2 weeks... now all you've got is 2 consecutive weeks of doing nothing but walking forward, of putting one foot in front of the other in hopes of reaching more trail to put one foot in front of the other on top of until you get to more trail and so forth. Of course, we all know that there's more to backpacking than this mundane description implies, but that's how your body can see it unless you have a strong sense of determination and can ignore that little voice in your head wants you to stop, that says this is far enough for today, and so forth. It's all about mental preparation.