So, will this problem arise? There are a couple of reasons for optimism: firstly, boredom is a human concept that we can avoid programming into AIs (however, such a concept may evolve within an AI outside of our control); secondly, my own theory of human boredom is that it arises from our inability to effectively multi-task. We cannot ring-fence the intellectual capacity needed for a certain task, and use the rest for something else, hence we allocate too much intellectual resource to a task that doesn't require it. An AI agent may however be able to do this ring-fencing, and as a result may not be bored.
(I'm sure my concerns are in part attributable to Douglas Adams' Marvin.)