  In this paper, we investigate which classes of monomial ideals have a
quasi-additive property of homological shift ideals. More precisely, for a
monomial ideal $I$ we are interested to find out whether $HS_{i+j}(I)\subseteq
HS_i(HS_j(I))$. It turns out that $\mathbf{c}$-bounded principal Borel ideals
as well as polymatroidal ideals satisfying strong exchange property, and
polymatroidal ideals generated in degree two have this quasi-additive property.
For squarefree Borel ideals, we even have equality. Besides, the inclusion
holds for every equigenerated Borel ideal and polymatroidal ideal when $j=1$.
