definitely does, a cold infection can protect the airways from a flu infection[1]. Cross reactivity of T cells very much exists. Another case is particular variants of covid-19 which is a corona virus granted partial immunity to influenza (one reason people suggested why both cold and flu seasons during covid peaks slowed considerably, apart from containment measures), and certain corona viruses that cause the common cold in turn also granted partial immunity against covid.
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.
Interesting comment on gluten consumption. In my case, it appears cumulative. (I'm not celiac, so far as I know.) I've been following a non-gluten diet for fifteen years at this point, and I can now tolerate at least a small amount of gluten without the effects I used to have.
I used to get injections to diminish my hay fever symptoms. I finally mostly grew out of hay fever around the age of forty, though I'm still allergic to horses and cats, if I have direct contact.
Don't those injections comprise increasing challenges to the immune system using the allergens themselves, to make it less prone to mount a histamine response (i.e. runny nose, itchy eyes)?
[1]https://news.yale.edu/2020/09/04/common-cold-combats-influen...