Yes they can
While Carebears has already said this, I'm going to shove my reasons in;
While we have no actual proof in the anime they are able to, since none of the challangers are shown or said to have rechallanged the league. However, it is a different case in the manga. It is said in the Manga that Professor Oak beat the League in his youth before.
n his youth, Professor Oak was a great Pokémon Trainer; he was strong enough to even defeat his old rival and friend, Agatha, and beat the Pokémon League
However in chapter 39, he is shown to have rechallanged the League and defeat Green before forfeiting for some reason.
Link to the manga chapter if you want
Additionally in the anime, in Ash's first match against Mandi in the Indigo League, Ash is called a league 'rookie' by the commentator, while Mandie is called the astounding bla bla bla. While this is not hard evidence in anyway, it does half imply Mandie has had experience in League matches before, so possibly competed before hand.
But logically speaking, there's also nothing against rechallanging the league. I mean Ash could rechallange the Battle Frontier alot, so why not the league? His just a lazy bum and can't be bothered.