yes. bronchits or any other inflammation in the body may cause fatigue, however other common problems such as poor or insufficient sleep as well as sleep apnea among other conditions may play a role

Unless the bronchitis was very severe and the patient elderly, excessive sleepiness should not occur after recovery. Even if the first situation applies, recovery should be complete within a week or so after the illness. Antibiotics in themselves are not associated with sleepiness/sedation.