The answer to the question depends on the timescale. The human response to sleep restriction is different, depending whether it is acute or chronic.
In the early days of sleep research, most experiments involving sleep restriction were total sleep deprivations, meaning participants in the study were allowed no sleep at all for some period of time, usually 1-3 nights (although a few crazy studies did go over 200 hours).
In these types of experiments, it was discovered that recovery tends to occur quite rapidly. To explain this, I need to refer to a model of human sleep regulation called the two-process model. In the two process model, it is assumed that human sleep is primarily regulated by two physiological processes: a circadian process and a sleep homeostatic process. The circadian process is the approximately 24-hour biological rhythm in sleepiness/alertness. The sleep homeostatic process is the process that promotes sleep more the longer that you have been awake.
Sleep homeostatic pressure builds up during time spent awake and dissipates during time spent asleep. Although we don't yet know precisely what causes the sleep homeostatic process (it may be the build up of sleep-promoting substances in the brain, including adenosine), it turns out that there is a good physiological marker for the sleep homeostatic process.
Normally, during a night of sleep, people cycle semi-regularly between states of NREM sleep and REM sleep. If you record somebody's brain electrical activity using EEG, you find that during NREM sleep, there is a high level of delta waves. These show up as big waves cycling about once per second in the EEG recordings. Across the night, the amount of delta waves in NREM sleep decreases approximately exponentially. Moreover, if you deprive someone of sleep and then let them get recovery sleep, their delta waves still decline exponentially across the night, but the initial level of delta waves at the beginning of the night is significantly higher.
It turns out that the two-process model can do a very good job of explaining the changes in delta waves across the night and in response to total sleep deprivation if you assume that the homeostatic sleep pressure builds up exponentially with a time constant of ~20 hours during wakefulness, and decays exponentially with a time constant of ~3 hours during sleep.
This has two important implications. First, the homeostatic sleep pressure would be expected to saturate to a maximum level quite rapidly -- within only a few days, given the time constant of ~20 hours. Second, even after a huge sleep deprivation, the homeostatic sleep pressure would be expected to return to approximately normal levels within a night or two of sleep, because the time constant for dissipation is only ~3 hours. Indeed, both of these predictions are true for total sleep deprivation. Even when people are deprived of sleep for 100+ hours, they tend to sleep for no more than about 14 hours on the first night of recovery. In other words, they do not pay back every hour of sleep missed.
For a while, it seemed like we therefore had sleep regulation figured out, at least in essence. However, the two-process model fails miserably when it is applied to the more common type of sleep restriction in day-to-day life: chronic sleep restriction for many consecutive days, e.g., getting only 6 hours of sleep per night for 14 consecutive days. Under these conditions, the two-process model would again predict that sleep homeostatic pressure would level off within a few days and recovery would be achieved at the end within one or two nights, i.e., a weekend. That is absolutely not what we see.
When individuals are chronically restricted of sleep for periods of 2-3 weeks, we instead find that cognitive impairment accumulates day by day, almost linearly. There is no sign of saturation or leveling off. Things just continue to get worse and worse. Paradoxically, delta waves do level off, just as the two-process model would predict, and so do subjective ratings of sleepiness, meaning people become less and less aware of their level of objective impairment as they are increasingly sleep restricted. After two weeks of getting 6 hours of sleep per night, individuals have the same reaction time as somebody who has been awake for 24 hours, which is approximately equivalent to an individual with a blood alcohol concentration of 0.10%. After two weeks of getting 4 hours of sleep per night, individuals have the same reaction time as somebody who has been awake for 48-72 hours.
The process of recovery also seems to be much slower after chronic sleep restriction, although it has not yet been well quantified. For chronic sleep restriction, there seems to be a much closer to one-to-one correspondence between hours of sleep lost and hours that must be paid back to return to baseline performance. Certainly, it is not possible to reverse the effects of chronic sleep restriction in a single weekend.
We don't yet know what is the physiological process underlying this much slower timescale response to chronic sleep restriction, but there are some hypotheses currently being tested, including up-regulation of adenosine receptors.
So, what about the effects of chronic sleep restriction on even longer timescales? What if you don't get enough sleep continually for a year, or a decade? How long would it then take you to recover? We don't know the answer to that. Laboratory studies of chronic sleep restriction cannot logistically or ethically go much longer than a month. We are therefore forced to rely on epidemiological data. We know that people who habitually get short sleep (less than about 6 hours) have higher rates of all-cause mortality, heart disease, obesity, and diabetes. But we don't know how easily those long-terms outcomes can be reversed by improving sleep patterns. For that, we will need longitudinal data, where people are tracked for years to see what happens if they improve their sleep habits. Those data would be very difficult to obtain, since it is difficult to convince people to make major lifestyle changes, and difficult to control for other lifestyle changes that may go along with them. Chronic sleep restriction also leads to increased hunger and poor diet choice, for instance, which may be one important confounding factor in such a study.
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