Next time you are on a plane, cruising over the ocean, coasting through clouds, marvelling at the miracle of air travel, consider this: it's possible both of your pilots are asleep. A survey of 500 commercial pilots has found that one in six has woken up on the flight deck to find their co-pilot snoozing.

Then, before you panic, consider this: it may actually be safer for them to nap. Even if, ideally, they should be doing it in shifts. The real danger for passengers, according to the British Airline Pilots Association (Balpa), is that our pilots are not sleeping enough.

"No aircraft in the history of aviation has crashed because a pilot has gone to sleep at the controls," says David Learmount, a former RAF Hercules pilot and safety editor at trade magazine Flight Global. "It's never happened. On the other hand, crashes that have resulted from fatigue? There are many, many, many of those."

Balpa commissioned the survey ahead of a European parliament vote on new EU regulations for pilots' flying hours. It argues that in certain circumstances, under the proposed new rules, pilots could be expected to land planes after 24 hours without sleep. Balpa say the resulting level of fatigue would be equivalent to being four times over a pilot's legal alcohol limit.

Half of the pilots surveyed identified tiredness as the single biggest threat to flight safety. By contrast, just 3% of them said the greatest threat was terrorism. More than half of them said they had fallen asleep on the flight deck. And 84% said their abilities had been compromised because of fatigue in the past six months.

"Fatigue," explains Learmount, "has an effect on anybody who suffers from it, the symptoms of which are remarkably like drunkenness. Your decisions are worse and your coordination is worse." A sleeping pilot is much less of a threat to passengers. "If the autopilot disconnects, the alarm is very loud. In a funny sort of way it's a bit safer than falling asleep on the motorway because there you have only got the rumble strip to wake you up."

Now consider this: one day in the not-so-distant future there might not even be a pilot onboard. Earlier this year, a 16-seater Jetstream plane completed the first unmanned flight across UK airspace: a 500-mile journey from Lancashire to Inverness. The plane, known as "the Flying Testbed", used robotics and onboard sensors to identify hazards and avoid them.

The test flight required a pilot on the ground to control the plane. But even they could one day be obsolete. In 2011, James Albaugh, then CEO of Boeing Commercial Airlines, told a conference of engineers: "A pilotless airliner is going to come. It's just a question of when."

So, on second thoughts, don't worry that your pilots might be getting 40 winks. Just be glad they're on the plane.

This article has been amended to add information about "the Flying Testbed" flight.