So, for most of us, eight hours of sleep is excellent and six hours is no good, but what about if we split the difference? What is the threshold below which cognitive function begins to flag? While Dinges’s study was under way, his colleague Gregory Belenky, then director of the division of neuroscience at the Walter Reed Army Institute of Research in Silver Spring, Md., was running a similar study. He purposely restricted his subjects to odd numbers of sleep hours — three, five, seven and nine hours — so that together the studies would offer a fuller picture of sleep-restriction. Belenky’s nine-hour subjects performed much like Dinges’s eight-hour ones. But in the seven-hour group, their response time on the P.V.T. slowed and continued to do so for three days, before stabilizing at lower levels than when they started. Americans average 6.9 hours on weeknights, according to the National Sleep Foundation. Which means that, whether we like it or not, we are not thinking as clearly as we could be.

Of course our lives are more stimulating than a sleep lab: we have coffee, bright lights, the social buzz of the office, all of which work as “countermeasures” to sleepiness. They can do the job for only so long, however. As Belenky, who now heads up the Sleep and Performance Research Center at Washington State University, Spokane, where Van Dongen is also a professor, told me about cognitive deficits: “You don’t see it the first day. But you do in five to seven days. Unless you’re doing work that doesn’t require much thought, you are trading time awake at the expense of performance.”

And it’s not clear that we can rely on weekends to make up for sleep deprivation. Dinges is now running a long-term sleep restriction and recovery study to see how many nights we need to erase our sleep debt. But past studies suggest that, at least in many cases, one night alone won’t do it.

Not every sleeper is the same, of course: Dinges has found that some people who need eight hours will immediately feel the wallop of one four-hour night, while other eight-hour sleepers can handle several four-hour nights before their performance deteriorates. (But deteriorate it will.) There is a small portion of the population — he estimates it at around 5 percent or even less — who, for what researchers think may be genetic reasons, can maintain their performance with five or fewer hours of sleep. (There is also a small percentage who require 9 or 10 hours.)

Still, while it’s tempting to believe we can train ourselves to be among the five-hour group — we can’t, Dinges says — or that we are naturally those five-hour sleepers, consider a key finding from Van Dongen and Dinges’s study: after just a few days, the four- and six-hour group reported that, yes, they were slightly sleepy. But they insisted they had adjusted to their new state. Even 14 days into the study, they said sleepiness was not affecting them. In fact, their performance had tanked. In other words, the sleep-deprived among us are lousy judges of our own sleep needs. We are not nearly as sharp as we think we are.