Consciousness is subjective experience. During both sleep and anesthesia, consciousness is common, evidenced by dreaming. A defining feature of dreaming is that, while conscious, we do not experience our environment; we are disconnected. Besides inducing behavioral unresponsiveness, a key goal of anesthesia is to prevent the experience of surgery (connected consciousness), by inducing either unconsciousness or disconnection of consciousness from the environment. Review of the isolated forearm technique demonstrates that consciousness, connectedness, and responsiveness uncouple during anesthesia; in clinical conditions, a median 37% of patients demonstrate connected consciousness. We describe potential neurobiological constructs that can explain this phenomenon: during light anesthesia the subcortical mechanisms subserving spontaneous behavioral responsiveness are disabled but information integration within the corticothalamic network continues to produce consciousness, and unperturbed norepinephrinergic signaling maintains connectedness. These concepts emphasize the need for developing anesthetic regimens and depth of anesthesia monitors that specifically target mechanisms of consciousness, connectedness, and responsiveness.