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After critiquing some earlier attempts (including those of Marcus Singer and Frances Howard–Snyder) to ground objections to actual–consequence act utilitarianism (ACAU) on human cognitive limitations, I present two new objections with this same foundation. Both start with the observation that, because human cognitive abilities are not up to the task of reliably recognizing utility–maximizing actions, any agents who are recognizably human – including the best possible humans, morally speaking – are certain to perform many actions every day that ACAU says are immoral, and to perform some actions over the course of their lives that it says are highly immoral. The first objection is that, if Mill's analysis of what it means to call an action wrong is accurate, then ACAU entails a conclusion that no one will accept, viz., that the morally–best humans possible ought to undergo constant punishment. The second objection is that ACAU entails that even the morally–best humans possible are bad moral agents in some respect. This conclusion, while unpalatable, is not so obviously unacceptable as the first. However, I do briefly consider some ways in which it might be possible to demonstrate that it is false and thereby complete a reductio of ACAU.