This is a continuation of discussion started in FINALLY: The Difference between Nerd, Dork, and Geek Explained by a Venn Diagram
Venn diagrams works only for two or three sets, but not for four because 4 circles in the plane do not divide the plane in 16 regions. Venn diagrams are useful only because axioms of Boolean algebra can be written using only three variables, thus allowing for a diagrammatic representation of each axiom. I have made a similar observation elsewhere: Coxeter groups allow a powerful level of visualisation because all relations between their canonical generators are in some explicit mathematical sense two dimensional.
A later addition in response to a comment: If we do not insist on sets being represented by circles, then Venn diagrams (although increasingly non-intuitive) can be drawn for larger numbers of sets:
However, their educational value is limited.
And this is a link to Branko Grunbaum.