As BBC Radio 4 reported the death of Senator Kennedy on the news, I heard a line about how his career had been blighted by the incident at the bridge at Chappaquiddick where "he failed immediately to report an accident". You can see what has happened: in an inadvisable attempt to avoid a split infinitive, the adverb has been placed before to, but this puts it next to failed, so we get interference from a distracting and unintended meaning that involves immediate failure (whatever that might mean). It was the reporting that should have been immediate. The right word order to pick would have been "he failed to immediately report an accident". But you just can't stop writers of news copy from being worried (falsely) that splitting an infinitive is some kind of mistake.
I should add one thing. Many of the comments that began appearing below the first version of this post (I have deleted them, for reasons I will now explain) wanted to argue for "he failed to report an accident immediately". But in fact that option was rendered disastrously less plausible in the BBC sentence, which I shortened (thinking the rest of the object irrelevant). It was roughly this: "he failed immediately to report an accident in which he drove off a bridge and his female companion drowned". It would of course be much worse to try "he failed to report an accident in which he drove off a bridge and his female companion drowned immediately", where almost no one would take the adverb to modify the verb report.
Other commenters want to defend "he failed to report immediately an accident". That would in fact also have been acceptable in the full context (which I did not originally give), but only because the noun phrase was long.
Still other commenters simply quibbled with earlier commenters or made sarcastic remarks about them or made the ridiculous suggestion that "an Americanism" was involved (Americans who find a sentence questionable always say it sounds British and British speakers say it's an Americanism — they blame whichever side of the Atlantic they know least about). These bored me, and I spiked them all.
Let me make this clear: I'm not saying that you never have a choice, and I'm not saying the split infinitive is always the right choice to make. All I'm saying that, squirm though you may, it is fairly common for placing an adverb between infinitival to and the following plain-form verb to be not just grammatical (it is always grammatical), but also the best stylistic choice. And this was one. But BBC editors resist that and worry about it. Stupidly.