  We give examples of $n$-sequentially compact spaces that are not
$(n+1)$-sequentially compact under several assumptions. We improve results from
Kubis and Szeptycki by building such examples from $\mathfrak{b=c}$ and
$\diamondsuit(\mathfrak{b})+\mathfrak{d}=\omega_1$. We also introduce a new
splitting-like cardinal invariant and then show that the same holds under
$\mathfrak{s=b}$.
