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It depends a lot on the genre and the expectations the author sets up for the reader. In a mystery or procedural where the plot logic is the main draw, inconsistencies are less forgivable, whereas a fantasy or speculative story can get away with a lot of hand-waving.
The easiest way that a lot of authors get away with inconsistency these days, though, is to set the story up as at least partly absurdist from the start. They might use an unreliable narrator, introduce some quantumy metaphysics, go for full blown magical realism, or take the meta-narrative approach of hanging a lantern on unlikely events by declaring them "like something that would happen in a novel." Going meta in a lighthearted way a la Gaiman can be very satisfying for contemporary readers while also giving the author all the wiggle room in the world short of total incoherence.
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