Everyone has "cancer or whatever" resistance. That's why DNA works, it has repair mechanisms.

Getting cancer is when that mechanism either fails or isn't good enough to repair the damage.

Abnormal radiation levels can cause an excess of damage or different type of damage than what your natural mechanism is capable of fixing.

We're constantly being radiated, we're constantly employing our resistances and defenses against radiation.

We float around on a rock in a sea of radiation and even we ourselves emit low levels of (mostly harmless) radiation.

And that trait does exist in nature already, it's just rare and mostly useless until environmental pressures only allow those individuals to reproduce.