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The NY times got it correct. A lot of other news outlets were reporting that cell phones cause cancer, based on prelimineray results of a study that found 2% of male rats and 0% of female rats exposed to a lot of cell phone radiation developed brain tumors. None of the rats in the control group developed tumors, but in the past, historically with this line of rats, 2% would be expected to develop brain tumors. That none did in the control group is a simple statistical fluke - it can easily happen that a few less or more than expected develop tumors. And that is what happened in the treatment group - the expected 2% of males developed it but the females didn't.
The Times pointed out that the incidence of brain tumors in the USA has not gone up over the last 20 years, during a time when cell phone use has grown exponentially.
My thoughts - there is no reason to give up your cell phones.
http://www.nytimes.com/2016/05/28/health/cancer-study-radiation-cellphones.html?_r=0