The rib cage and sternum are part of the chest. Yes, a feeling of tightness when one decides to breathe more deeply or does so with exercise, is a common experience with asthma. However, there are other causes of chest "tightness", including musculoskeletal tightness from the rib cage itself and tightness due to heart disease, especially with exertion.
You should have your chest examined, get a chest x-ray and arrange to have spirometry, a simple breathing test.
If you are a young person, less than 30 years old, with a history of asthma, this would be far and away the most likely cause of the tightness.
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