The sequential activation of neurons, reflecting a previously experienced temporal sequence of stimuli, is be- lieved to be a hallmark of learning across cortical areas1, including the primary visual cortex2,3 (V1). While circuit mechanisms of sequence learning have been studied extensively4,5, the converse problem, that is equally important for robust performance, has so far received much less attention: how to avoid producing spurious sequential activity that does not reflect actual sequences in the input? Here, we developed a new measure of sequentiality for multivariate time series, and a theory that allowed us to predict the sequen- tiality of the output of a recurrent neural circuit. Our theory suggested that avoiding spurious sequential activity is non-trivial for neural circuits: e.g. even with a completely non-sequential input and perfectly sym- metric synaptic weights, the output of a neural circuit will still be sequential in general. We then show that the most celebrated principles of synaptic organization, those of Hebb and Dale, jointly act to effectively pre- vent spurious sequences. We tested the prediction that cortical circuits actively diminish sequential activity, in an experience-dependent way, in multielectrode recordings from awake ferret V1. We found that activity in response to natural stimuli, to which animals were continually adapted, was largely non-sequential. In contrast, when animals were shown entirely non-sequential artificial stimuli, to which they had not been adapted yet, neural activity was sequential at first, and then gradually became non-sequential within a few minutes of extended exposure. Furthermore, this difference between responses to natural and artificial stimuli was not present at eye opening but developed over several days. Our work identifies fundamental requirements for the reliable learning of temporal information, and reveals new functional roles for Dale’s principle and Hebbian experience-dependent plasticity in neural circuit self-organization.