Three weeks before giving birth, 34-year-old Kyla Bryan went to the hospital with shortness of breath and dizziness—symptoms the doctors decided were pregnancy-related and nothing to be overly concerned about.

In February, eight weeks after her daughter was born, she experienced terrible stomach pains, could barely breathe when she was lying down nursing, and had chest pains when she carried her baby up the stairs. At the emergency room, an echocardiogram showed that her heart was pumping significantly less blood than normal,...