This paper takes a novel approach to a known basic difficulty with computer simulations of nonlinear dynamical systems relevant to climate modeling. Specifically, we show by minimal examples how small systematic modeling errors might survive averaging over an ensemble of initial conditions. The resulting predictive errors can grow slowly enough initially that they may be overlooked without contradicting known behaviors on middle scales. However, they may nonetheless be significant on long timescales, given our current knowledge. Mathematical symmetry, which has been investigated for improving accuracy in computational algorithms, turns out to provide a novel perspective to this issue.