Changes in the activity profile of cortical neurons are due to effects at the scale of local and long-range networks. Accordingly, abrupt transitions in the state of cortical neurons---a phenomenon known as Up-Down states---have been attributed to variation in the activity of afferent neurons. However, cellular physiology and morphology may also play a role in causing Up-Down states. This study examines the impact of dendritic nonlinearities, particularly those mediated by voltage-dependent NMDA receptors, on the response of cortical neurons to balanced excitatory/inhibitory synaptic inputs. Using a neuron model with two segregated dendritic compartments, we compared cells with and without dendritic nonlinearities. NMDA receptors boosted somatic firing in the balanced condition and increased the correlation between membrane potentials across the compartments of the neuron model. Dendritic nonlinearities elicited strong bimodality in the distribution of the somatic potential when the cell was driven with cortical-like input. Moreover, dendritic nonlinearities could detect small input fluctuations and lead to Up-Down states whose statistics and dynamics closely resemble electrophysiological data. Up-Down states also occurred in recurrent networks with oscillatory firing activity, as in anaesthetized animal models, when dendritic NMDA receptors were partially disabled. These findings suggest that there is a dissociation between cellular and network-level features that could both contribute to the emergence of Up-Down states. Our study highlights the complex interplay between dendritic integration and activity-driven dynamics in the origin of cortical bistability.