We analyze dissipative boundary conditions for nonlinear hyperbolic systems in one space dimension. We show that a known sufficient condition for exponential stability with respect to the H-2-norm is not sufficient for the exponential stability with respect to the C-1-norm. Hence, due to the nonlinearity, even in the case of classical solutions, the exponential stability depends strongly on the norm considered. We also give a new sufficient condition for the exponential stability with respect to the W-2,W-p-norm. The methods used are inspired from the theory of the linear time-delay systems and incorporate the characteristic method.