As an extension of prior work, we studied inspecific Hebbian learning using the classical Oja model. We used a combination of analytical tools and numerical simulations to investigate how the effects of synaptic cross talk (which we also refer to as synaptic inspecificity) depend on the input statistics. We investigated a variety of patterns that appear in dimensions higher than two (and classified them based on covariance type and input bias). We found that the effects of cross talk on learning dynamics and outcome is highly dependent on the input statistics and that cross talk may lead in some cases to catastrophic effects on learning or development. Arbitrarily small levels of cross talk are able to trigger bifurcations in learning dynamics, or bring the system in close enough proximity to a critical state, to make the effects indistinguishable from a real bifurcation. We also investigated how cross talk behaves toward unbiased (“competitive”) inputs and in which circumstances it can help the system productively resolve the competition. Finally, we discuss the idea that sophisticated neocortical learning requires accurate synaptic updates (similar to polynucleotide copying, which requires highly accurate replication). Since it is unlikely that the brain can completely eliminate cross talk, we support the proposal that is uses a neural mechanism that “proofreads” the accuracy of the updates, much as DNA proofreading lowers copying error rate.