We prove that any set of points in $\mathbb{R}^d$, any three of which form an angle less than $\fracπ{3} + c$, has size $(1+Θ(c))^d$ for sufficiently small $c>0$. The proof is based on a refinement of an approach by Erdős and Füredi. The lower bound is relying on a problem about large hypegraphs with small edge intersections, while the upper bound is tightly connected to the problem of packing disjoint caps on a sphere.