I think the examples given are instances of Guy's "strong law of small numbers". That seems at least poetic reason for low-dimensional specializations of your favorite theory to be different in character from high-dimensional specializations.
An example of increasing richness indicating Thom was thinking about something else:
- "The" connected 0-manifold is a point
- "The" connected compact 1-manifold is a circle
- Connected compact 2-manifolds are connect sums of tori or of projective planes; they are uniformizable.
- Connected compact smooth 3-manifolds are piecewise geometrizable, where the joints are among spheres and tori.
- Connected compact $(3+n)$-manifolds "solve" the word problem for groups; particularly weird: there are smooth examples that have contractible stable closed periodic geodesics in any smooth metric.
This is sort-of what I'd call rigidity in low dimensions, and richness in high dimensions. Perhaps the theories in low dimension are richer in the sense that there are more universal statements we can prove, but there seems to be a greater wealth of useful examples in higher dimension.
Whether this is an instance of Thom's principle as quoted or an exception, it is still an instance of Guy's law, in that the low-dimensional behaviour isn't representative of high-dimensional behaviour.