In odorant-rich environments, animals must be able to detect specific odorants of interest against variable backgrounds. However, several studies have suggested that both humans and rodents are very poor at analyzing the components of odorant mixtures, leading to the idea that olfaction is a synthetic sense in which mixtures are perceived holistically. We have developed a behavioral task to directly measure the ability of mice to perceive mixture components and found that mice can be easily trained to detect target odorants embedded in unpredictable and variable mixtures. We imaged the responses of olfactory bulb glomeruli to the individual odors used in the task in mice expressing the Ca++ indicator GCaMP3 in olfactory receptor neurons. By relating behavioral performance to the glomerular response patterns, we found that the difficulty of segregating the target from the background was strongly dependent on the extent of overlap between the representations of the target and the background odors by olfactory receptors. Our study indicates that the olfactory system has powerful analytic abilities that are constrained by the limits of combinatorial neural representation of odorants at the level of the olfactory receptors.