Does gaining weight mean you will get taller?

Does not mean that, no.

It depends on your metabolism. It also takes time to grow and it depends on what foods you are eating. If it's all unhealthy like process foods and fast foods it is most likely going to fat. If you are eating healthy foods such meats, veggies, fruit, dairy, it should be helping in many parts of the body but as stated it takes time to grow. At this age, growing starts to slow down unless you haven't reached your growth spurt. It would probably be best to see your percentage of body fat because if you aren't looking 'fat' after gaining twenty pounds, it probably went to muscle mass or somewhere else to support your body. Your twenty pounds gained might just be muscle. Muscle weighs more than fat. If you truly want to figure out. Start recording your waist measurement and your height often. But to simply answer the question. Gaining weight doesn't always mean getting taller especially if there is a great weight difference. Sorry for the long answer