Everyday that I go to work, I walk around the outside of the aircraft. That walkaround is essentially to check that all the various bits of the airplane are where they are supposed to be. Many of those "bits" are radio aerials. Some look like sharks fins, some are round, some are flat. They come in a variety of shapes and sizes on the various parts of the fuselage and empannages. However they are all carefully built into the physical and electrical structure of the aircraft, so that they work efficiently, and also so that they don't interfere with each other.
Once the passengers board, another 300 odd radio aerials are much less precisely introduced. Given the "cage effect" of an aircraft fuselage, those little radio boxes usually need to up their reception and transmission strength in order to overcome the "cage" obstruction.
Many is the time, when we have to deal with electrical glitches that may or may not be contributed to by the operation of these devices, so for the sake of this one illustration, let's ignore them.
One thing that these phones do without any doubt, is cause interference to the aircrafts radios. That interference manifests itself in a way that is obvious and will be familiar to anybody who has ever tried speaking on a landline telephone with a cellphone placed nearby, or nearby the wiring.
That DA.DA.DA.DA acquisition signal that makes communication difficult, and is solved by moving the cellphone away. When that happens in the pilots headphones (as it regularly does) it can and sometimes does, result in mis-heard and blocked communications. That really is something that is not good. Communication degradation can be a frequent nuisance from other sources as well, but this is one that can be largely eliminated if the laid down rules are properly complied with.
The worst disaster in civil aviation history (los Rodeos 1977,) was in significant part down to communication failure, including hetrodynes that blocked out ATC communications. People often get excited about Ipads taking over sophisticated electronic control systems. In truth the real threat is much more mundane, and one that anyone can experience for themselves, just by leaving their iphone next to their bedside landline tonight.