  We study conditions under which sub-complexes of a double complex of vector
spaces allow to compute the Bott-Chern cohomology. We are especially aimed at
studying the Bott-Chern cohomology of special classes of solvmanifolds, namely,
complex parallelizable solvmanifolds and solvmanifolds of splitting type. More
precisely, we can construct explicit finite-dimensional double complexes that
allow to compute the Bott-Chern cohomology of compact quotients of complex Lie
groups, respectively, of some Lie groups of the type
$\mathbb{C}^n\ltimes_\varphi N$ where $N$ is nilpotent. As an application, we
compute the Bott-Chern cohomology of the complex parallelizable Nakamura
manifold and of the completely-solvable Nakamura manifold. In particular, the
latter shows that the property of satisfying the
$\partial\overline\partial$-Lemma is not strongly-closed under deformations of
the complex structure.
