Using free stochastic calculus we give a lower estimate on the free entropy dimension for Bozejko-Speicher q-semicircular families, which is valid for small values of q. Combining this with earlier results of Voiculescu, Ge, Stefan, Jung and others, this gives certain structural information about the von Neumann algebra generated by these families; in particular, for small q, these algebras have no Cartan subalgebras.