We prove a purely combinatorial obstruction for the Bloch-Kato property within the class of fundamental groups of complement manifolds of toric arrangements (i.e., arrangements of hypersurfaces in the complex torus). As a stepping stone we obtain a combinatorial obstruction for the cohomology of a supersolvable arrangement to be generated in degree 1. Our result allows us to prove that - for all prime numbers $p$, the pro-$p$ completion of the pure braid group on $k$ strands has the Bloch-Kato property if and only if $k\leq 3$; - for all prime numbers $p$, the pro-$p$ completion of the pure mapping class group of the sphere $S^2$ with $k$ punctures has the Bloch-Kato property if and only if $k\leq 4$.