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Note: I'm going to be talking about productivity in the sense of hard work. Eg. Anders Ericsson's idea of deliberate practice.

It seems self-evident that if you're performing busywork like data entry or cleaning, you could be productive for way longer than four hours/day.

It also seems self-evident that in doing something in between busywork and hard work, you can still be productive for more than four hours/day. For example, a calculus student doing tons of practice problems, or an experienced web developer building a simple sign up form.

So then, it seems to make sense to me to deal with the question of hard work. Pushing yourself to the edge of your comfort zone, or perhaps slightly beyond it. For example, mastering a difficult new concept or skill.

Of course, when we talk about hard vs easy work, we're talking about a spectrum, so "hard work" really refers to a general area in that spectrum, not a single point. The answer to the question of how many productive hours you have in a day probably depends a little bit on where exactly you are on that spectrum, but getting into that nuance seems like something more appropriate for a different question. Here, it seems like it's worth just grouping "hard work" together and treating it as a single thing.

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Note: As jimrandomh mentions, things like age, physical health and motivation probably matter, as do things like performance enhancing substances or techniques. It seems to me like it'd make sense to address those things in different question though. Eg. here I think we can ask the question of how many productive hours a healthy individual who isn't doing any performance enhancing stuff has in a day. Then in a separate question, we can ask how things like age and physical health impact this number. And in another separate question, we can ask how things like performance enhancing substances impact the number.

Answer

After about an hour of googling around, I haven't been able to find any of those high quality studies either. My impression is that the "people are only productive for four hours/day" idea largely comes from interviewing successful people, as opposed to rigorously measuring their performance. If there were high quality studies that really measured performance, I'd expect that they would be talked about more and easier to find, so I take the fact that they're hard to find as reasonably strong evidence that they don't really exist.

It is interesting to note that across a wide range of experts, including athletes, novelists, and musicians, very few appear to be able to engage in more than four or five hours of high concentration and deliberate practice at a time.

This quote comes from a Harvard Business Review article titled The Making of an Expert. The authors are K. Anders Ericsson, Michael J. Prietula, and Edward T. Cokely. I know that Ericsson is known as an authority in the study of expert performance; not sure about the other authors.

The article Why you should work 4 hours a day, according to science also had some relevant excerpts. The author mentions various examples of experts reporting that four hours/day is their limit, and that in interviewing people, this is the strong pattern.

Darwin:

After his morning walk and breakfast, Charles Darwin was in his study by 8 a.m. and worked a steady hour and a half. At 9:30 he would read the morning mail and write letters. At 10:30, Darwin returned to more serious work, sometimes moving to his aviary or greenhouse to conduct experiments. By noon, he would declare, "I've done a good day's work," and set out on a long walk. When he returned after an hour or more, Darwin had lunch and answered more letters. At 3 p.m. he would retire for a nap; an hour later he would arise, take another walk, then return to his study until 5:30, when he would join his wife and family for dinner.

Poincaré:

Toulouse noted that Poincaré kept very regular hours. He did his hardest thinking between 10 a.m. and noon, and again between 5 and 7 in the afternoon. The 19th century's most towering mathematical genius worked just enough to get his mind around a problem — about four hours a day.

G.H. Hardy:

G.H. Hardy, one of Britain's leading mathematicians in the first half of the 20th century, would start his day with a leisurely breakfast and close reading of the cricket scores, then from 9 to 1 would be immersed in mathematics. After lunch he would be out again, walking and playing tennis. "Four hours' creative work a day is about the limit for a mathematician," he told his friend and fellow Oxford professor C.P. Snow.

John Edensor Littlewood:

Hardy's longtime collaborator John Edensor Littlewood believed that the "close concentration" required to do serious work meant that a mathematician could work "four hours a day or at most five, with breaks about every hour (for walks perhaps)."

Ericsson's study of violin students:

Add these several practices up, and what do you get? About four hours a day. About the same amount of time Darwin spent every day doing his hardest work and Hardy and Littlewood spent doing math.

This upper limit, Ericsson concluded, is defined "not by available time, but by available [mental and physical] resources for effortful practice."

The article also makes some more general statements about the four hours/day idea:

Darwin is not the only famous scientist who combined a lifelong dedication to science with apparently short working hours. We can see similar patterns in many others' careers

In particular, it mentions the following study:

A survey of scientists' working lives conducted in the early 1950s yielded results in a similar range. Illinois Institute of Technology psychology professors Raymond Van Zelst and Willard Kerr surveyed their colleagues about their work habits and schedules, then graphed the number of hours spent in the office against the number of articles they produced. You might expect that the result would be a straight line showing that the more hours scientists worked, the more articles they published. But it wasn't.

The data revealed an M-shaped curve. The curve rose steeply at first and peaked at between 10 to 20 hours per week. The curve then turned downward. Scientists who spent 25 hours in the workplace were no more productive than those who spent five. Scientists working 35 hours a week were half as productive as their 20-hours-a-week colleagues.

If you search through Cal Newport's blog, you'll find lots of other examples of expert performers only being able to manage four hours/day of hard work. John Grisham’s 15-Hour Workweek is one:

Grisham primarily writes his novels during the winter months on his farm in Oxford, Mississippi. During this period he works five days a week, starting at 7 am and typically ending by 10 am.

There is something important that I'd like to emphasize. From what I can tell, all of these examples of expert performers who can only perform about four hours of hard work per day, these people all seem to be getting the other parts of the productivity equation right. They're sleeping enough, taking naps, taking walks, eliminating distractions, quitting Facebook, etc. So four hours/day seems like it is conditional on those things. I think that this is worth noting because many of us are not getting those things right. I know I'm not.