This headline is a good example of why this rule is a bad practice in English:
Officials: Sen. Hagel not to run again
By DAVID ESPO
AP SPECIAL CORRESPONDENT
WASHINGTON -- Nebraska Sen. Chuck Hagel, a persistent Republican critic of the Iraq war, intends to announce on Monday he will not seek a third term, according to Republican officials.
He's "not to run" again? Does that mean he's been forbidden to run? That's what it sounds like.
But this isn't the case; the article says later it was his own decision.
A better solution here would be to say "won't run" instead of "not to run." But "to not run again" would have at least kept the intended meaning.
So go ahead and split infinitives if it keeps your meaning intact. Keep the words together in other cases, just so you won't raise the dander of the nuns and other rigid constructionists.