Anyone that has flown in the last two decades has certainly heard those words, but few (airline employees included) actually know why. Perhaps a cell phone will inadvertently run the plane into the ground, or someone's GameBoy will get it's wires crossed with those in the cockpit. The truth, it turns out, is far less interesting. But this doesn't stop the myths and rumors from spreading.

Simple answer: no.
The not-so-simple answer: well...still no. There is some truth to the myth; aircraft do use radio communication extensively for navigation. In addition to the audio communication between pilots and controllers, planes also follow radio navigation beacons. These beacons (VHF Omnidirectional Radio-Ranges, or VORs) emit constant radio signals to help planes navigate from point to point over the ground. While a scrambled signal here might send a plane in the direction of Mexico instead of Montana, it is unlikely that any interference would cause the dramatic crash so popular in people's minds.
The more sensitive system at risk is the ILS, or Instrument Landing System. The ILS is a system that allows planes to come in for a landing in bad weather, or when the runway is otherwise not visible to pilots. It emits a very focused beam at a set radio frequency, and pilots can use detecting equipment in the plane to "ride" this beam down to the runway for landing. Anyone that's seen the movie Die-Hard Two will recognize this system as the one that crashed Miles O'Brien's plane. By shifting the glideslope down, the bad guys were able to trick the plane into flying itself into the ground. This sort of mess is the feared potential of cell phone interference, but even so, it is very unlikely, if not impossible.
The fact of the matter is that cell phones and aircraft systems operate at very different frequencies. The "Airband" containing all navigation and communication radio frequencies falls between 108 and 136MHz. The lowest cell phone frequency in use is at 806MHz, and climb as high as 4G at 2690MHz.

So, you may be asking "then why have the airlines banned cell phone use?" Well, they haven't. The current ban on cell phone use in flight was put into effect by the Federal Communications Commission (yes, that FCC...you now have another reason to hate them) in 1991. The fear was not damage to airplanes, but damage to the Cell Phone networks themselves. A cell phone at 35,000 feet and traveling 500 miles per hour will bounce between cell phone towers on the ground, trying to connect to multiple towers at once, and using up much more capacity than any normal phone call. Afraid that this practice in any large scale would interfere with the nation's regular communications, the FCC established the ban on in-flight use.

No. Not a chance. If it doesn't transmit any radio signals, it won't be interfering with anything.
But before you start popping open your laptop on takeoff, you should know that there's a reason airlines make you put your "approved" electronics away.

Takeoff or landing is the most dangerous phase of any flight. Imagine your seatmate makes a cell phone call at the wrong time, and the plane smashes into a retaining wall at the end of the runway. Picture the guy sitting behind you...large touristy type, eager to to jet off to Orlando, with a hefty camcorder in his hand to record the takeoff. While it might not be the first thing on your mind, that camera probably won't agree with the back of your head when they meet because of the forces of the crash.
Okay, okay, that's rather unlikely, but take a much more common example. Imagine the wing outside your window bursts into flames while you're still on the ground. Being buried in your laptop and music, you don't see or hear any of the commotion. You remain seated and happy playing Halo, while the plane burns around you. Grim, yes. But slightly more realistic than you may think.

Situational awareness is key in the event of an aircraft emergency, and being alert and un-encumbered by electronics may be key to getting out of an emergency safely. I admit, I rarely follow the "unapproved electronics" rule to the letter (this is where I hide the dozens of photos and videos I've taken on takeoff and landing...), but it is there for the safety of the passengers.

Surprisingly, yes, it may.
Any equipment that generates a radio signal that comes close to those used by the plane can cause interference, and even a little bit of interference can be deadly.
In March of 1977, on the island of Tenerife in the Canary Islands, a KLM 747 prepared for takeoff. At the same time, half way down the fog-shrouded runway, a PanAm 747 taxied straight towards it. The captain of the KLM aircraft, impatient to take off began to spool up the engines and go. At the same moment, the PanAm Captain and the Air Traffic Controller tried to stop the KLM plane in its path. Unfortunately, the towers words of "stand by for takeoff" came at the exact same time as the other 747's "PanAm, still taxing down the runway." Instead of being transmitted to the KLM captain and halting his takeoff, both transmissions were garbled on the frequency, in what is called a heterodyne. When two radio communications overlap on a VHF frequency, all that can be heard is a high pitched squeal, as the two transmissions interfere with one another. In Tenerife, this led to two fully laden 747s crashing into one another, the deadliest air disaster in the world to date. Radio procedures prevent this from happening today, but if another source were to transmit illegally on the air-band, the potential for a disastrous miscommunication still exists.

So next time you go flying, leave your RC transmitters at home, turn off your cell phones, and enjoy the flight. The chance of dangerous interference is slim to none, but without extensive testing, it's impossible to know what would happen if 200 people all started transmitting different radio signals inside an aluminum tube all at once. It's better to play it safe, and not piss off the FCC at the same time.