A recent but rapidly maturing held in the area of system identification has been that of 'estimation in H-infinity'. Greatly influencing this work has been the phenomenon that no linear (in-the-data) algorithm exists which is 'robustly convergent'. This paper conducts a study of this issue by combining specific new analysis together with existing results from the mathematics literature on the topic of polynomial approximation theory. Particular attention is paid to the role of model order, and this leads to the consideration of model order selection from a deterministic worst-case perspective.