  We prove that the dimension $h^{1,1}_{\overline\partial}$ of the space of
Dolbeault harmonic $(1,1)$-forms is not necessarily always equal to $b^-$ on a
compact almost complex 4-manifold endowed with an almost Hermitian metric which
is not locally conformally almost K\"ahler. Indeed, we provide examples of non
integrable, non locally conformally almost K\"ahler, almost Hermitian
structures on compact 4-manifolds with $h^{1,1}_{\overline\partial}=b^-+1$.
This answers to a question by Holt.
