This article is concerned with the norm-inflation phenomena associated with a periodic initial-value abcd-Benjamin-Bona-Mahony type Boussinesq system. We show that the initial-value problem is ill-posed in the periodic Sobolev spaces Hp−s(0,2π)×Hp−s(0,2π) for all s>0. Our proof is constructive, in the sense that we provide smooth initial data that generates solutions arbitrarily large in Hp−s(0,2π)×Hp−s(0,2π)-norm for arbitrarily short time. This result is sharp since in [13] the well-posedness is proved to holding for all positive periodic Sobolev indexes of the form Hps(0,2π)×Hps(0,2π), including s=0.