I've processed as late as 20-22 weeks and still had good meat.  I keep the meat birds more confined, in tractors or small pens.  I'd even give them recliner chairs, big-screen TVs and remote controls so they'd be more sedentary and less tough, but they'd only just poop on it all.
So many folks have told me about how tough or stringy their home-grown chickens have been but I haven't encountered that yet.  Perhaps it's because I keep them confined, or let the carcasses rest in the refrigerator a few days.  I also usually slow-simmer them over low heat for a few hours on the stove with some fresh herbs, then pick & separate the meat to freeze.  I don't have to really pick, the meat just melts off the bones.
I guess I should start weighing these birds to see if they really gain much between weeks 16-22, perhaps it's not worth it to keep, feed, & listen to them for so much longer.  But it's good to know that if you're too busy to butcher so early that doing it later shouldn't affect the taste/texture so much.