When looking at it from an overall perspective, bulldogs have been known to have no more health problems than many an other bred of dog. Bulldogs can actually experience things like epilepsy, cancer, arthritis, even something that is known as canine diabetes, though in dogs this one particular trait is a bit different, they are most similar to humans than any other breed.
Still, the health matters being described are more canine specific, than humanoid. Sometimes when the dog seems that it is simply unhealthy, one may need to take a closer look.
Link: Bulldog Health and Preventive Medicine