"Gawker writer J.K. Trotter published a report on Monday that alleged Fox News host Bill O'Reilly was accused during divorce proceedings of assaulting his ex-wife before they separated."
I laughed when I first read this sentence because I initially parsed it as calling Bill O'Reilly an alleged Fox News host rather than the intended meaning, in which alleged is a verb. This ambiguity could have been avoided if the writer had inserted that in front of the content clause ("that alleged that..."). Of course then there would be two thats in quick succession.
The obvious way around that stylistic infelicity is to change the first that to which. It's at this point, however, that an American copy-editing fetish interferes. Many American editors and teachers favor using that for all restrictive relative clauses and which for nonrestrictive clauses, with no good reasons to justify their preference.
The obvious and natural way to rewrite this sentence is "published a report on Monday which alleged that Fox News host...," and while I have no idea what the actual sequence of composing was, it seems at least quite plausible that the writer started with which and then either self-corrected or had an editor revise it without carefully considering the resulting ambiguity.
The silly rules that are so often propounded over that and which contribute nothing to the clarity of English. More often, they give rise to ambiguity. If you're a teacher who insists on this precept in your students' papers, please stop. You're hurting your students' writing. Most editors are probably stuck enforcing house rules, regardless of whether or not they approve of them, but it wouldn't hurt if they campaigned for a change.
Bonus point: this sentence also illustrates a right-dislocated relative clause. That is, the relative clause "(which) alleged (that)..." is separated from its antecedent by an intervening phrase ("on Monday"). This too violates another precept commonly taught in writing classes: always put relative clauses next to their antecedents. But there's a good chance you never even noticed that dislocation, because it doesn't create any real stylistic problems. English favors putting heavy clauses like this relative clause towards the right edge of the sentence.