Already for $n=3$ the answer is no. Indeed, $h^{3,3}=1$ so by your condition $h^{1,1}=h^{2,2}=0$ but a compact Kähler manifold has $h^{1,1}>0$.

EDIT: this answer refers to a previous version of the question.

Already for $n=3$ the answer is no. Indeed, $h^{3,3}=1$ so by your condition $h^{1,1}=h^{2,2}=0$ but a compact Kähler manifold has $h^{1,1}>0$.