In most information-theoretic studies of communication, it is assumed that the channel is always available. However communication between two parties may cease due to damage to transmitter/receiver devices, wire lines being cut, loss of attention at the terminals, or other such reasons. To model a setting where attention must be maintained, here we consider a channel model - A boring channel - with a reserve of attention that degrades when the channel is being used but is replenished when a codeword containing a certain amount of Bayesian surprise is received. Limits of communication over the boring channel are characterized and numerical results are found. Optimal communication strategies may be useful for various cognitive systems, and coincide with the segmenting principle that states that students learn better from a multimedia lesson when it is presented in user-paced segments rather than as a continuous unit. An application to designing the interaction between people and artificial agents is also discussed.