  We introduce a "qualitative property" for Bott-Chern cohomology of complex
non-K\"ahler manifolds, which is motivated in view of the study of the
algebraic structure of Bott-Chern cohomology. We prove that such a property
characterizes the validity of the $\partial\overline\partial$-Lemma. This
follows from a quantitative study of Bott-Chern cohomology. In this context, we
also prove a new bound on the dimension of the Bott-Chern cohomology in terms
of the Hodge numbers. We also give a generalization of this upper bound, with
applications to symplectic cohomologies.
