Cold & flu weren't the best examples. Here's a more obvious one: getting food poisoning doesn't make you less likely to catch the flu.

Some people treat "the immune system" like a muscle, where stress strengthens. Similar stressors can have cross-reactive effects, but unrelated ones do not.

This can go the other way too: people who have uncontrolled celiac disease (as in, they're still consuming gluten on a regular basis) can also have issues with avenin (a somewhat similar protein in oats) or even lactose. For a lot of folks those sensitivities disappear when they get their gluten consumption down.