  In this paper, we consider the Cauchy problem for $(abcd)$-Boussinesq system
posed on one- and two-dimensional Euclidean spaces. This model, initially
introduced by Bona, Chen, and Saut, describes a small-amplitude waves on the
surface of an inviscid fluid, and derived as a first-order approximation of
incompressible, irrotational Euler equations. We mainly establish the
ill-posedness of the system under various parameter regimes, which generalize
the result of the one-dimensional BBM-BBM case by Chen and Liu. Most of results
established here, we obtain the optimal result for two-dimensional BBM-BBM
system. The proof follows from an observation of the \emph{high to
low-frequency cascade} present in nonlinearity, motivated by Bejenaru and Tao.
