We present new results on dynamical instabilities in rapidly rotating relativistic stars. In particular, using numerical simulations in full general relativity, we analyse the effects that the stellar compactness has on the threshold for the onset of the dynamical bar-mode instability, as well as on the appearance of other dynamical instabilities. By using an extrapolation technique developed and tested in our previous study (Baiotti L et al 2007 Phys. Rev. D 75 044023), we explicitly determine the threshold for a wide range of compactnesses using four sequences of models of constant baryonic mass comprising a total of 59 stellar models. Our calculation of the threshold is in good agreement with the Newtonian prediction and improves the previous post-Newtonian estimates. In addition, we find that for stars with sufficiently large mass and compactness, the m = 3 deformation is the fastest growing one. For all of the models considered, the non-axisymmetric instability is suppressed on a dynamical timescale with an m = 1 deformation dominating the final stages of the instability. These results, together with those presented in Baiotti L et al (2007 Phys. Rev. D 75 044023), suggest that an m = 1 deformation represents a general and late-time feature of non-axisymmetric dynamical instabilities both in full general relativity and in Newtonian gravity.