Despite the dearth of field-based evidence from natural model-mimic communities, theory suggests that Batesian mimicry should have limits placed upon the model:mimic ratio for mimics to benefit. Paradoxically, hoverflies that are apparently mimics are often superabundant, many times more abundant than their supposed models. One possible solution to this paradox is that perhaps they are not mimics at all. We use discriminative operant conditioning methods to measure the similarity perceived by pigeons between wasps and various species of supposedly mimetic hoverflies, and an image processing technique to measure objective similarity. We demonstrate that pigeons rank mimics according to their similarity to a wasp model, in an orderly broadly similar to our own intuitive rankings. Thus pigeons behave as if many hoverflies are indeed wasp mimics. However, they rank the two commonest hoverflies as very similar to wasps, despite these looking decidedly poor mimics to the human eye. In these species, `poor' mimicry may have been sustainable because it exploits some constraint in birds' visual or learning mechanisms, or some key feature used in pattern recognition. Furthermore, the relation between similarity and mimicry is nonlinear: small changes in similarity can lead to dramatic increases in the degree of mimicry.