A recent study published in the Journal of Experimental Psychology found that the effects of placebo sleep (being told you've gotten quality sleep) are enough for the body to feel the benefits of actual sleep. Like Peter Pan and the Lost Boys who simply imagined they had food, and all of the sudden they were full.
Researchers from Colorado College tested the effects of "placebo sleep" on the most poorly rested demographic, undergraduates. They hooked participants up to equipment that measured brainwave frequency, then told the students that the measurements tell how much REM sleep they'd received the night before. They lied. And students who were told they had above-average sleep performed better on the test researchers administered, while those who were told they had poor quality sleep performed worse.
There's been plenty of focus on the dire health consequences of sleeping badly: depression, weight gain, anxiety, organ failure, green scales, etc. Thanks, science, for giving us this work-around.