Sensory systems can maintain perceptual constancy over stimulus intensities spanning many orders of magnitude. In the olfactory system, where odour identity is thought to be represented by a combinatorial code, a cascade of mechanisms is proposed to maintain object constancy in the face of concentration dependent changes in the activated receptor channels. In this study we explore the neurobiology at the boundary condition of where an odour percept breaks down with concentration. We use a combination of behavioural paradigms and in vivo imaging of neural activity within the olfactory bulb and find that a change in odour percept corresponds to rapid and complete adaptation in just a few glomeruli that are most sensitive to the odour. This concentration dependent shift to fast adaptation is already present in the signal delivered by the olfactory receptor neurons and is not a property of circuit interactions within the olfactory bulb. It therefore seems that mechanisms to promote concentration invariance are unable to operate when rapid peripheral adaption has occurred. Furthermore, our data imply that odour identity relies on a sparse combinatorial code, as perceptual constancy depends on the activity of only a few olfactory receptor channels.