I always understood cancer to be caused by a flaw in division during mitosis (by whatever means). This is when the DNA is most vulnerable. If the random flaw happens to be in a place in the DNA that alters the code to increase reproduction and/or remove the self-destruct component, the result becomes a cancer cell.

With that said... you would need much lower levels of radiation to cause cancer over a long period of time.

This is how radiation treatment and chemotherapy work against cancer as well. Cancerous cells reproduce faster than normal cells, so when exposed to the toxin (chemo drugs are literally poison) or radiation, you're killing the cancer cells faster than the healthy ones (though doing damage to both).