Why would the sleeping brain spend more REM time after a hard day of learning?
The idea behind these studies is that REM sleep does some of the work of handling new information. If that were true, a period of unusually intense learning should leave more material to process, and the brain might respond by spending more time in REM sleep afterward. Researchers have used this logic to test the hypothesis directly: load up on new material, then measure the sleep that follows [1].
The evidence is consistent with that expectation. After an intensive second-language course, REM sleep increased [1]. After a motor learning task, the group that learned had significantly more REM sleep than the comparison group [2]. And after an intensive learning period, people showed increases in the number of rapid eye movements (REMs) and in REM density, which is how closely packed those eye movements are during REM sleep [3]. These are different measures of the same sleep stage, and they moved in the same direction.
It is worth being precise about what this does and does not mean. The studies show a change in REM sleep after learning; they do not show that REM sleep is what caused the learning to stick. The proposed role of REM sleep in processing new information is a hypothesis these studies were designed to test, not a conclusion they establish [1].
Does a bigger REM increase mean someone learned more?
This is the part of the question the classic studies largely leave open. The second-language study examined changes in REM sleep and dream content but did not provide evidence on whether the REM increase tracked individual learning outcomes [1]. The motor learning study demonstrated an increase in REM sleep after learning but did not explicitly assess whether the size of that increase correlated with how well each person performed [2]. The study of REMs and REM density likewise did not explore whether the extent of those increases related to individual learning outcomes [3].
So for the most direct version of your question, the honest answer is that these studies were not built to answer it. They compared learning conditions or looked at change over time, not the match between each person's REM change and each person's learning score.
One study did look at REM sleep alongside individual learning performance, but in a very different setting: military veterans with posttraumatic stress disorder doing safety signal learning, a form of learning about when a situation is safe. There, learning was associated with less efficient REM sleep the following night, and fragmented REM sleep was associated with safety learning performance [4]. That is not a simple story of "more REM, better learning." It suggests the relationship may depend on the kind of learning and the kind of sleeper, and it comes from a preliminary study in a clinical group, so it should not be generalized to healthy learners without caution [4].
Could something other than learning explain the extra REM sleep?
A rise in REM sleep after a demanding task is not automatically proof that the task's content drove it. The review of sleep, learning, and memory raises a concrete alternative: stress. It notes that future research should rule out the role of stress on subsequent REM sleep, and points out that some animals that learned a task still did not show a REM sleep increase [5]. That combination matters. If learning reliably produced extra REM sleep, you would expect learners to show it consistently; the exceptions suggest other factors are in play.
This is also why the question of individual differences is hard to answer cleanly. If stress, task type, and the person's baseline sleep all influence REM sleep, then a single number like "minutes of REM" may mix several influences together. The studies that found increases [1][2][3] establish that something about intensive learning periods is followed by more REM sleep, but they do not isolate which ingredient is responsible.
What should you take away if you are asking about your own learning?
The most defensible summary is this: intensive learning periods have repeatedly been followed by increases in REM sleep and in REM-related measures such as eye movement count and density [1][2][3]. That pattern is real in the studies that looked for it. What is not established is a dose-response link, where learning more produces a proportionally larger REM increase. The studies that found increases generally did not test that link [1][2][3], and the one study that connected REM measures to individual learning performance found an inverse-leaning pattern in a PTSD sample rather than a straightforward boost [4].
If you are trying to use sleep to support learning, the evidence supports treating REM sleep as part of the picture but not as a scoreboard. A night with more REM sleep after a hard study session is consistent with the brain doing something with new material, but it is not a reliable signal that the material was learned well, and a smaller REM increase is not evidence that learning failed.
Sources used in this answer
The effects of intensive learning of a second language on REM sleep and dream content
Intensive second-language learning was followed by an increase in REM sleep, but the study did not test whether the size of that increase tracked individual learning outcomes.
Effect of a motor learning task on REM sleep parameters
A group that performed a motor learning task had significantly more REM sleep than a comparison group, but the study did not assess whether the increase correlated with individual learning performance.
Increases in number of REMS and REM density in humans following an intensive learning period
Following an intensive learning period, humans showed significant increases in the number of rapid eye movements and in REM density, without analysis of how these increases related to individual learning outcomes.
REM sleep and safety signal learning in posttraumatic stress disorder: A preliminary study in military veterans
In military veterans with PTSD, safety signal learning was associated with less efficient REM sleep the following night, and fragmented REM sleep was associated with safety learning performance.
The role of sleep in learning and memory
The review discusses REM sleep increases in relation to learning but notes that some learning animals did not show a REM sleep increase and that future research should rule out the role of stress on subsequent REM sleep.