We consider the Cauchy problem associated to the generalized Benjamin-Bona-Mahony (BBM) equation for given data in the L2-based Sobolev spaces. Depending on the order of nonlinearity and dispersion, we prove that the Cauchy problem is ill-posed for data with lower order Sobolev regularity. We also prove that, in certain range of the Sobolev regularity, even if the solution exists globally in time, it fails to be smooth.341145654576Alazman, A.A., Albert, J.P., Bona, J.L., Chen, M., Wu, J., Comparisons between the BBM equation and a Boussinesq system (2006) Adv. Differential Equations, 11, pp. 121-166Angulo Pava, J., Banquet, C., Scialom, M., Stability for the moDiffed and fourth Benjamin- Bona-Mahony equations (2011) Discrete Contin. Dyn. Syst., 30, pp. 851-871Benjamin, T.B., Bona, J.L., Mahony, J.J., Model equations for long waves in nonlinear dispersive systems (1972) Phil. Trans. Royal Soc. London, 272, pp. 47-78Bona, J.L., Pritchard, W.G., Scott, L.R., An evaluation of a model equation for water waves (1981) Philos. Trans. Royal Soc. London Series A, 302, pp. 457-510Bona, J.L., Tzvetkov, N., Sharp well-posedness results for the BBM equation (2009) Discrete and Continuous Dynamical Systems, 23, pp. 1241-1252Bona, J., Chen, H., Well-posedness for regularized nonlinear dispersive wave equations (2009) Disc. Cont. Dynamical Systems, 23, pp. 1253-1275Bourgain, J., Periodic Korteweg de Vries equation with measures as initial data (1997) Selecta Math. New Ser., 3, pp. 115-159Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation (1993) Geom. Funct. Anal., 3, pp. 209-262Chen, W., Li, J., On the low regularity of the moDiffed Korteweg-de Vries equation with a dissipative term (2007) J. Diff. Equations, 240, pp. 125-144Christ, M., Colliander, J., Tao, T., Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations (2003) American Journal of Mathematics, 125 (6), pp. 1235-1293Molinet, L., A note on the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space, arXiv:1110.2352v1 (2013) Proc. Amer. Math. Soc., 141, pp. 2793-2798Molinet, L., Ribaud, F., On the low regularity of the Korteweg-de Vries-Burgers Equation (2002) Int. Math. Research Notices, pp. 1979-2005Molinet, L., Ribaud, F., Youssfi, A., Ill-posedness issues for a class of parabolic equations (2002) Royal Society of Edinburgh - Proceedings A, 132 (6), pp. 1407-1416Molinet, L., Saut, J.-C., Tzvetkov, N., Ill-posedness issues for the Benjamin-Ono and related equations (2001) SIAM. J. Math. Anal., 33, pp. 982-988Panthee, M., On the ill-posedness result for the BBM equation (2011) Discrete Contin. Dyn. Syst., 30, pp. 253-259Tzvetkov, N., Remark on the local ill-posedness for KdV equation (1999) C. R. Acad. Sci. Paris Ser. i Math., 329, pp. 1043-1047