We study the dynamics of entanglement in the scaling limit of the Ising spin chain in the presence of both a longitudinal and a transverse field. We present analytical results for the quench of the longitudinal field in the critical transverse field which go beyond current lattice integrability techniques. We test these results against a numerical simulation on the corresponding lattice model finding extremely good agreement. We show that the presence of bound states in the spectrum of the field theory leads to oscillations in the entanglement entropy and suppresses its linear growth on the time scales accessible to numerical simulations. For small quenches, we exactly determine these oscillatory contributions and demonstrate that their presence follows from symmetry arguments. For the quench of the transverse field at zero longitudinal field, we prove that the Renyi entropies are exactly proportional to the logarithm of the exponential of a time-dependent function, whose leading large-time behavior is linear, hence, entanglement grows linearly. We conclude that, in the scaling limit, linear growth and oscillations in the entanglement entropies can not be simply seen as consequences of integrability and its breaking, respectively.