One in five pilots suffers from fatigue in the cockpit at least once a week, a study seen by the BBC suggests.
The study, commissioned by the British pilots' union, Balpa, and carried out by University College London (UCL), is due to be released on Thursday.
Balpa is worried the situation will get worse if European proposals regarding flying hours come into force.
But European authorities claim the proposed changes will not lead to a reduction in safety.
The study by UCL sent questionnaires to pilots working for a major British airline.
A total of 492 replied, which marked a 47% response.
Some 45% of those said they were suffering from significant fatigue. One in five reported their abilities were compromised in flight more than once a week.
'I fell asleep'
One pilot who works for another large UK airline, but who did not want to be named, told the BBC that, about three months ago, both he and his co-pilot had very little sleep during their rest period.
On the subsequent flight, his co-pilot asked if he could take a nap, which the pilot approved. But then the pilot fell asleep too - for about 10 minutes.
"When I woke up, it was a big adrenalin rush. The first thing you do obviously is check your height and your speeds and all of your instrumentation," he said.
"The worst scenario is that the autopilot would disconnect itself and then the aircraft would lose or gain height and that would be extremely dangerous as you'd go into the path of oncoming aircraft.
"Now there are warning systems that tell you you are deviating from the correct altitude but they are not excessively loud - it would be easy enough to sleep through that, and I probably don't need to tell you what the consequences of that are."
Failure to report
More than three-quarters of pilots who returned questionnaires admitted there were times within the past six months when they had been tired but had not filed a fatigue report with their airline.
About 19% of those who did not file a report when tired said it was because they did not want management to have a less positive opinion of them. But 6% said they did not file a report because the fatigue was their own fault.
The report also notes that those who chose to reply might have been biased towards reporting higher levels of fatigue in order to avoid the potential threat of new, more relaxed regulations.
Balpa, which commissioned the report, is warning that problems with fatigue could get worse under European proposals that would see the maximum flying time for UK pilots going up from 900 hours in a 12-month period to 1,000 hours.
"The problem with fatigue is that it slows your reactions down. When you are travelling towards the ground at 200mph and you have to make a decision within one or two seconds, if you don't make that right decision, that can lead to disaster," said Dave Smith from Balpa.
"The landing can be 16 hours after the take-off and while the passengers have been asleep, we have to stay awake and we have to be at our alertest at the end of those 16 hours. That's a big task."
At the moment, the rules for UK airlines are set by the regulator and are some of the toughest in the world. But airlines from other European countries operate under their own rules.
Not only that, but the rise of low-cost airlines has changed the way the industry works. For example, their crews will do far more take-offs and landings within their hours than pilots on long-haul flights, who spend more time cruising on autopilot.
So the European authorities want to standardise regulations across the continent.
Balpa would like the new regulations to be set at the tougher, UK level. But other countries would be likely to object, saying that would be too much regulation and would damage their airlines.
So, in practice, the new rules are expected to relax the regulations for UK airlines, but make them much tougher for other airlines. That will benefit UK travellers on airlines with weaker rules.
"The change in the European regulations will mean some countries will raise their standards for the passenger which will of course include UK passengers travelling on those airlines," said Captain Bob Jones, head of flight operations for the Civil Aviation Authority (CAA).
The CAA is happy with some aspects of the European proposals but wants changes in other areas.
However, the European Aviation Safety Agency argues its proposed rules are just as safe as the current UK ones.
It is now considering the views of Balpa and the CAA before putting the final proposals before the Council of Ministers.
When the resulting proposals are approved, individual countries will have no choice but to adopt the measures.
Correction 9 April 2011: Two figures in this report were amended following clarification of some of the survey's findings.