Isn't it abnormal for your chest to feel tight with your asthma?

Not normal.: Normal only if you are put into a vise. Tightness means you have trouble with airway conduction. You need to see your doctor because you end up with a terrible attack. In the meanwhile, you may feel tired often as well even in the absence of shortness of breath. There are also other factors which can cause chest tightness and your doctor may need to do some other tests as well.