I've bought into that illusion in the past

Catching up on lost sleep a dangerous illusion

People who are chronically sleep-deprived may think they're caught up after a 10-hour night of sleep, but new research shows that although they're near-normal when they awake, their ability to function deteriorates markedly as night falls.

Some studies show that almost 30% of Americans get less than six hours of sleep at night. The research indicates that the body's daily circadian rhythm hides the effects of chronic sleep loss and gives such people a second wind between about 3 p.m. and 7 p.m., when the circadian rhythm is pushing them to be awake.

But then they fall off a cliff in terms of attention.

According to the article, staying up for 24 hours can impair a person on par with being over the legal limit for alcohol. Getting only 6 hours of sleep for two weeks gives about the same results as being up for 24 hours. Since many professions where attention is crucial involve little sleep--paramedics, doctors, nurses firemen, policemen, truckers, etc.--critical errors may result. Scary.