Could you get shortness of breath due to menstrual cycle?

Not sure: Well, this is not an uncommon complaint. I doubt if it is related to the bleeding itself as if you are anemic that should be present throughout the cycle. Of course, it probably is related to hormone levels which are lowest during the bleeding. Doctors at the cleveland clinic discovered that blood flow to the uterus and lungs increases during the first 2 weeks of the cycle so that may be pertinent.

Not usually: There are some lung problems however which can vary with the menstrual cycle. One of these is asthma. Some women with asthma find their symptoms worsen around the time of menses because hormonal changes may affect the disease. A much rare problem is a fluid collection that can occur in the space around the lung with menses in those with endometriosis. Shortness of breath requires a doctor visit.

Very rarely: Asthma flares have been reported with menses, but this is very rare. Endometriosis with spread to the chest cavity is also possible - but this is associated with pain and (occasionally) spontaneous pneumothoraces.