When I exercise, my chest gets really tight. I don't think this is an endurance problem. Could this be a lung problem?

Possibly. Chest tightness with exercise can be from a number of causes, the most serious being heart or lung problems. If one experiences tightness, an immediate evaluation by your physician is warranted. Exercise should be stopped until this evaluation occurs.
Or heart problem. Should consult with your doctor and possibly get a stress echocardiogram and pulmonary function tests (with diffusion capacity, dlco). The stress echo can be done with a treadmill or with medications. If normal, can potentially proceed with a cardiopulmonary exercise test to check out your lungs by measuring maximal oxygen consumption.