The study reported increased phone-side brain activity, raising questions about whether this can, over time, damage the brain. Whether this holds up, or is in any way associated with the fear that phoning causes cancer, we can't say. Whether this study is scientifically sound as well as newsworthy is a similar question.
Our statistical methods rely on sample sizes large enough to detect effects that are 'unusual' enough to take a serious evidence for cause. That is, large effects. But 'unusual' enough, otherwise known as 'statistically significant', is a totally subjective judgment. Something can look unusual by chance, and something truly very unusual (a perfect bridge hand, or all cherries on a slot machine) can seem to be fore-ordained, if you cannot do enough tests to prove that it was just what is expected by chance.
More disturbing is that something that is very rare, but very real can go undetected by our inferential methods. The cause may be so weak that no adequate sample could be collected for the outcome to be statistically significant by the usual kinds of 'unusual enough' criteria. But if it happens to you, it can kill you, and that's real enough to take seriously!
Similarly, suppose some group--say, chatty teenagers who talk on their mobiles in class--has a 1% risk of some brain disease (on top of not belonging in the class in the first place). Does that mean that each person has a mere 1% risk, small enough to ignore compared to the thrill of chatting up your favored co-ed? Or does it mean that 1 person in 100 has a 100% risk, and the others, in fact, can talk all they want with complete impunity?
There is no obvious easy way to get out of this box, to know what is 'real', other than what we declare to be real after we've run our Statistica program and got a p-value out of a significance test. That may be our currently preferred way, but it's a poor way to try to understand the real nature of Nature.