There are a number of reasons that you're asked to turn off your electronics for those phases of flight – and interference with the instrumentation is only one of them.
Mobile phones and devices equipped with Wi-Fi (like a Kindle or iPad) are the chief offenders. I have personally had mobile phones interfere with my headset a number of times... it creates a kind of buzzing sound that comes in spurts when the device is searching for a signal. It's annoying and has caused me to not hear a controller's instructions a couple of times.
The IATA (International Air Transport Association) published a report in which they observed 75 cases in the past few years where interference was suspected. Again, these were not test conditions... but the crew reported issues with all kinds of systems, made an announcement that people needed to shut stuff off, and then the problem went away.
The other reason the airlines want you to shut off your electronic gadgetry is that it's a distraction during a phase of flight where they might need your attention. Headphones interrupt what you might hear during an emergency, hand-held games have a way of absorbing people's attention, etc. Take the US Airways flight that landed in the Hudson, for example. That whole ordeal lasted only a few minutes, and the crew wouldn't have had time to repeat themselves to people who weren't listening.
So those are the reasons from a logical point of view. From a legal point of view, you're required to comply with the crew's instructions – failure to do so violates a federal law (found in part 14 of the Code of Federal Regulations). They don't often seek punishment for violations of this reg, but it has happened.
John Fiscus, chief pilot, flight instructor at The Flight Academy
Here's some great quotes from airline pilots about the topic:
"We don't make you stow your laptop because we're worried about electronic interference. It's about having a projectile on your lap. I don't know about you, but I don't want to get hit in the head by a MacBook going 200 miles per hour." – Patrick Smith
"People don't understand why they can't use their mobile phones. Well, what can happen is 12 people will decide to call someone just before landing, and I can get a false reading on my instruments saying that we are higher than we really are." – Jim Tilmon
"We're not trying to ruin your fun by making you take off your headphones. We just want you to be able to hear us if there's an emergency." – Patrick Smith
Katie Shackelford, pilot
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