  We study Bott-Chern and Aeppli cohomologies of a vector space endowed with
two anti-commuting endomorphisms whose square is zero. In particular, we prove
an inequality \`a la Fr\"olicher relating the dimensions of the Bott-Chern and
Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove
that the equality in such an inequality \`a la Fr\"olicher characterizes the
validity of the so-called cohomological property of satisfying the
$\partial\overline{\partial}$-Lemma. As an application, we study cohomological
properties of compact either complex, or symplectic, or, more in general,
generalized-complex manifolds.
