A nonlinear car-following model that includes the reaction-time delay of drivers is considered. When investigating the linear stability of the uniform flow solution, boundaries of Hopf bifurcations are determined in the parameter space. Crossing these boundaries, oscillations may appear corresponding to travelling wave solutions. Hopf normal form calculations prove robustly subcritical behavior which leads to bistability between the stable uniform traffic flow and the stop-and-go waves travelling against the flow of vehicles. Analogies with wheel shimmy dynamics and machine tool vibrations are presented.