  We study restrictions on cohomology algebras of Kaehler compact manifolds,
not depending on the h^{p,q} numbers or the symplectic structure. To illustrate
the effectiveness of these restrictions, we give a number of examples of
compact symplectic manifolds satisfying the Lefschetz property but not having
the cohomology algebra of a compact Kaehler manifold. We also prove the
stability of these restrictions under products.
