Using the concept of $s$-formality we are able to extend the bounds of a Theorem of Miller and show that a compact $k$-connected $(4k+3)$- or $(4k+4)$-manifold with $b_{k+1}=1$ is formal. We study $k$-connected $n$-manifolds, $n=4k+3, 4k+4$, with a hard Lefschetz-like property and prove that in this case if $b_{k+1}=2$, then the manifold is formal, while, in $4k+3$-dimensions, if $b_{k+1}=3$ all Massey products vanish. We finish with examples inspired by symplectic geometry and manifolds with special holonomy.