We have studied the cosmological evolution of domain wall networks in two, three and four spatial dimensions using high-resolution field theory simulations.The large number and size of our simulations has allowed us to measure scaling exponents with considerably smaller error bars. In all cases we find deviations to the expected scale-invariant evolution. Our results are in fact consistent with most previous numerical simulations, but these have typically neglected these or attributed them to numerical inaccuracies. Here we describe simulations suggesting that such deviations may well be real, as well as various other numerical and statistical tests to our code and results. However we point out that one is always limited by the finite dynamical range of our simulations and, as our results with the largest dynamical range show, it is possible that future simulations with a much larger dynamical range will produce results consistent with scaling. If this t! urns out to be the case, our simulations already reveal that the approach to the scaling will necessarily be slow. In addiction to the usual 2D and 3D numerical simulations, we have also performed some simulations of domain wall evolution in four spatial dimension. These can be relevant, in a phenomenological way, for brane-world scenarios. Finally, we point out that if the violations to scale invariance are real there will be significant modifications to cosmological scenarios involving domain walls.