Results true in a dimension and false for higher dimensions

Keller's conjecture asserts that whenever one tiles ${\bf R}^n$ by unit cubes, there must be two cubes which share a common face. True when $n \leq 6$, false for $n\geq 8$, and still open for $n=7$.

UPDATE, October 2019: It appears that the conjecture has now been resolved in the affirmative by computer assisted proof in $n=7$: https://arxiv.org/abs/1910.03740

An n-dimensional brownian motion visits every neighborhood of $\mathbb{R}^n$ infinitely often with probability 1 iff $n \leq 2$

Claim: Every smooth $n$-dimensional manifold homeomorphic to a sphere is also diffeomorphic to a sphere. In other words, there are no exotic spheres of dimension $n$.

True for $n=1,2,3,5,6,12,61$. Open for $n=4$. False for all other $n < 126$, and for all odd $n \geq 126$ (according to forthcoming work of Behren, Hill, Hopkins, and Mahowald, plus earlier results; there are some additional results for large even $n$ also, but I don't know the precise statements). The $n=7$ case was a famous counterexample of Milnor. The problem is closely connected to that of determining the (stable) homotopy groups of spheres, see e.g. the previous MathOverflow question Exotic spheres and stable homotopy in all large dimensions? .