That is not how radiation generally works.
If it was true, then the sun would case more random genetic effects.
This thought comes from movies like the incredible hulk.
Most often when we are exposed to ionizing radiation (Food, The Sun, Radon, Smoking, Pollution) our bodies repair any damage that occurs.
When our bodies can't repair the damage it can result in cell death or cancer.
The mutations common in movies are playing on the fear of radiation, but the reality is our bodies are amazing at catching errors in the genetic code and repairing them with enzymes.
The reality is, if mutations like this occurred then sunburns or household Radon would radically change physical appearances of people giving them third arms or the such.
Most chronic radiation exposure (low levels over a long time) does not have any more significant effect than other lifestyle choices (diet, exercise, chemical exposure, etc).
Acute (high levels over a short period) are extremely rare. You would need something like 1,000 dental x rays to meet your annual dosage.
In the US the limit is 50 mSv per year and a dental x ray is 0.005 mSV.
Most people get 1.5 - 3.5 mSv per year with background radiation (about a 1/3 due to Radon).
You would have to do something really stupid to be acutely exposed to radiation or piss off the Russian government.