We present fully atomistic molecular dynamics simulations on polyisobutylene (PIB) in a wide temperature range above the glass transition. The cell is validated by direct comparison of magnitudes computed from the simulation and measured by neutron scattering on protonated samples reported in previous works. Once the reliability of the simulation is assured, we exploit the information in the atomic trajectories to characterize the dynamics of the different kinds of atoms in PIB. All of them, including main-chain carbons, show a crossover from Gaussian to non-Gaussian behavior in the intermediate scattering function that can be described in terms of the anomalous jump diffusion model. The full characterization of the methyl-group hydrogen motions requires accounting for rotational motions. We show that the usually assumed statistically independence of rotational and segmental motions fails in this case. We apply the rotational rate distribution model to correlation functions calculated for the relative positions of methyl-group hydrogens with respect to the carbon atom at which they are linked. The contributions to the vibrational density of states are also discussed. We conclude that methyl-group rotations are coupled with the main-chain dynamics. Finally, we revise in the light of the simulations the hypothesis and conclusions made in previously reported neutron scattering investigations on protonated samples trying to address the origin of the dielectric β-process.