In one and two dimensions, transport coefficients may diverge in the thermodynamic limit due to long-time correlation of the corresponding currents. The effective asymptotic behavior is addressed with reference to the problem of heat transport in one-dimensional crystals, modeled by chains of classical nonlinear oscillators. Extensive accurate equilibrium and nonequilibrium numerical simulations confirm that the finite-size thermal conductivity diverges with system size L as kappa proportional to L alpha. However, the exponent alpha deviates systematically from the theoretical prediction alpha=1/3 proposed in a recent paper [O. Narayan and S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002)].