Is it possible for the acute bronchitis to still not go away even after antibiotics?

Bronchitis: Unfortunately, what you state is very possible. Many cases of bronchitis are caused by viruses, which are not treatable by antibiotics. If your bronchitis is not improving, you could either be on the wrong antibiotic, or your bronchitis is viral and not bacterial. If it's viral, it must just run it's course and rely on your body's natural defense mechanisms for a cure.

Yes: In a non-smoking, otherwise healthy 19 year old, bronchitis is hardly ever caused by a bacteria, so antibiotics have no effect. Most cases are viral and go away on their own. If it is not better after a couple of weeks, consider another diagnosis - usually it's asthma, sometimes whooping cough.