Quick note for future self: Here's study that was testing number-of-variables one could compare (here on SciHub, for now).

Abstract (emphasis mine)

The conceptual complexity of problems was manipulated to probe the limits of human information processing capacity. Participants were asked to interpret graphically displayed statistical interactions. In such problems, all independent variables need to be considered together, so that decomposition into smaller subtasks is constrained, and thus the order of the interaction directly determines conceptual complexity.

As the order of the interaction increases, the number of variables increases. Results showed a significant decline in accuracy and speed of solution from three-way to four-way interactions. Furthermore, performance on a five-way interaction was at chance level. These findings suggest that a structure defined on four variables is at the limit of human processing capacity.

Method

Participants

To optimize expertise for the task, we recruited 30 participants who were academic staff and graduate students in psychology and computer science and had experience in interpreting the type of data presented.

Materials

Each problem involved selecting the correct form of a verbal description to match a graphical representation of an interaction based on fictitious data on one of six everyday topics (see Figs. 1 and 2). The verbal descriptions were written so as to suggest that the lowest level of difference between pairs be treated as a single entity, such as a preference or difference (e.g., "People prefer fresh cakes to frozen cakes"). The sentences then described how the other variables in the interaction affected that preference entity (e.g., "The difference between fresh and frozen increases from chocolate cakes to carrot cakes"). Thus, the construction of the sentences encouraged conceptual constructions as described in the representational analysis just presented, in which a difference between two levels of one variable is operated on by further variables. To keep the complexity of the tasks equal at all levels of interaction, we used only binary variables.

The materials were designed to equalize all task characteristics except for complexity. In the crucial comparisons, the input memory load was kept as constant as possible by equalizing the amount of verbal information and the number of bars. Thus, 2 two-way interactions were compared with a three-way interaction using 8 bars (as shown in Figs, la and lb) and 2 three-way interactions were compared with a four-w

Related but not quite the same thing: because of the general inability to visualize or see things beyond 3D (and usually only 2D because making 3D graphs is a pain, especially in print), we resort to various tricks to examine high dimensional data in lower dimensions. This includes making multiple 2D, two-variable plots comparing only the interactions of pairs out of larger set of variables (obviously we lose out on more complicated relationship that way), and projecting higher dimensional stuff onto lower dimensions (PCA being often used to do this). Probably other techniques too, but the two above are quite common.

This tutorial does a good job describing the general idea of the above paragraph plus the two techniques described.

Arguably these are external tools which are being used to get around the limitations of human visualization/working memory capacity, tools which are more powerful than just having paper store plain information for you. These are much worse than if you could see in higher dimensions, but they're better than nothing.

Possibly there are generalizations to these kinds of techniques which could work for more general working-memory constraints? Probably they're not researched enough.