This paper considers the conditional fault tolerance, $h$-super connectivity $κ^{h}$ and $h$-super edge-connectivity $λ^{h}$ of the hierarchical cubic network $HCN_n$, an attractive alternative network to the hypercube, and shows $κ^h(HCN_n)=λ^h(HCN_n)=2^h(n+1-h)$ for any $h$ with $0\leq h\leq n-1$. The results imply that at least $2^h(n+1-h)$ vertices or edges have to be removed from $HCN_n$ to make it disconnected with no vertices of degree less than $h$, and generalize some known results.