I reckon if you go look at BBQ forums or facebook groups just about every second or third post is a deflated bbqer posting this desperate question after a wasted day and a disappointing dinner. There isn't a worse feeling in BBQ and we've all been there.
Depending on the cut your cooking the problem is usually one of a few things to lookout for.
If you're cooking a big cut like a pork shoulder or brisket or the answer is completely counterintuitive for a BBQ newbie. the reason your meat is chewier than a day old well done steak is actually because it's UNDERcooked. The key to beautiful BBQ is rendered connective tissue and intermuscular fat. Even if your brisket is up to temp it's likely that it needs to go for longer or has been brought up to temp too quickly. The difficult thing here is that every brisket/shoulder/shin has slightly different ratios of meat, fat and connective tissue so, there's unfortunately no step by step instructional that will work EVERY time. The best way to fix this is wrap her up with some liquid. beef stock and some tallow is great for brisket, I love apple juice and butter for pork. And, let her ride. Allow yourself a few hours grace in every cook (3-4 Hours) to keep it going if needed then use the extra time to rest.
For smaller or leaner cuts like chicken or ribs. Watch your internal temps you can definitely easily over cook these. Also watch any harsher direct heat this will over cook parts of a rack of ribs very quickly before the tissue has broken down even at lower temps. Keep it low and slow and watch for any hot spots in your cooking chamber.
Good luck BBQers!