It depends on the dosage. In medium to large doses that's certainly true, but it's not a settled question for low doses. The model that assumes that any radiation (no matter how little) causes damage is called the Linear No-Threshold model, and there's actually not a lot of scientific consensus on its validity.
It's a conservative model, and thus it tends to get used out of an abundance of caution, but it's hard to quantify low-dose effects accurately. Since we're all exposed to a fair bit of radiation naturally, low doses don't represent much change in total radiation exposure.
Average dosage to the public in the affected countries was about 9 millisieverts [1]. A CT scan is about 10 mSv. The average natural exposure is about 2.4 mSv per year in America, but in some places with high radon levels exposure can be up to 7 mSv/year. It's difficult to reach significance with small changes like that.
