In extreme cases, that can manifest as autoimmune disease, when overly strong inflammation or other immune responses end up attacking not just foreign pathogens but the person's body itself. As another poster said, inflammation is a blunt instrument. It's a knob that can only be turned up or down, across the entire body. If you turn it down too far, you risk infectious illness. And if you turn it up too far, you risk damage to your organs.
Interestingly, there was a substantial increase in the incidence of autoimmune diseases in Europe in the generations following the Black Death, probably because people with excessively strong immune responses were more likely to survive exposure to plague bacteria. Celiacs or MS will kill someone much, much more slowly than bubonic plague will, so a disproportionate number of people with those or similar autoimmune disorders were able to survive to pass on their genes.