Mythbusters took this one on in Episode 60, first aired in the US 15 March 2006.
Here's what happened.
1. Inside a Faraday cage, they blasted a simulated cockpit with mobile phone waves and managed to get the VOR to deflect when they hit the right frequency (800+ mhz), but it took a serious blast.
2. They took the simulated cockpit to the airport on the back of a truck and tried the same thing, but couldn't get the VOR to settle in the first place. An engineer said this was probably because the simulated cockpit didn't include any shielding.
3. The FAA in the US wouldn't even allow them to experiment in the air, so they got a Lear jet on the tarmac at the airport with a mobile phone wave generator blasting away in the back. No so much as one instrument in the Lear even flickered.
Conclusion: mobile phones do nothing to aircraft avionics because of the shielding. It was concluded that mobiles are banned because the FAA don't want to take any chances.
Myth busted.
Walrus