The whole point of cooking chicken is to kill off pathogenic bacteria.

There's literally no way properly-cooked chicken could "infect" raw chicken.

You can definitely get germs on already-cooked chicken, though, especially if you leave raw chicken/chicken juice on surfaces in your kitchen and cross-contaminate the cooked food. And since it's already warm (presumably cooling down from its cooking temp once it's off the heat) and not likely to be re-cooked to a high enough temp, any pathogens can reproduce quickly to toxic levels if you leave it out... someone could get sick from eating the cooked chicken. That's why you're supposed to quickly refrigerate leftovers of high-risk foods like meat - to inhibit pathogen growth.

You could conduct an experiment:

1. Take two boneless chicken breasts of equal weight and thickness.
2. Cook one to the recommended internal temp of 165F.
3. Put them both in ziploc bags and immerse in a tepid water bath for ~15 min to bring their internal temps close.
4. Leave the sealed bags on the counter, checking for bad smells, etc, every day until one of them goes bad.
5. If you have nerdy friends with a biology lab, you could even see what kind of microbes they can culture/identify from the samples.
6. Throw them BOTH in the trash when you're done. This shouldn't have to be said, but the way you asked that question...