Turns out that etymology experts are uncertain. The Oxford English Dictionary (OED) notes that hunters have sometimes been called nimrods as an allusion to the biblical hunter, and sometimes incompetent hunters have facetiously been called "nimrod," starting with a literary reference in 1933.
It may be Bugs Bunny that popularized this ironic use of "nimrod" (although the OED doesn't mention it). Bugs was known to occasionally call Elmer "poor little nimrod." The biblically illiterate audience was probably unaware of the allusion and picked it up as an insult for an incompetent person.
The pictures: above, Sargon I of Assyria, whom some identify with Nimrod; below, Elmer Fudd of Warner Brothers, whom Bugs Bunny identifies as "nimrod."