My mother was a child in Poland during the Chernobyl disaster. In her family nobody has a history of cancer but she had breast cancer. And a lot of other health issues. Coincidence? Maybe.
Fact is, we use epidemiology to understand these things, and the jury is in [1].
[1] https://www.unscear.org/unscear/en/chernobyl.html
Anecdotally, my college girlfriend was a child in Gomel Belarus during Chernobyl. Her father had a Geiger counter and took readings around here. There was lots of elevated dose rates. She is perfectly healthy to this day though!