Are there any anime that have English language dubs voiced by the voice actors of the original non-English language of the anime?
The most prominent example that comes to mind is Pokémon, which has used Ikue Ohtani's voice for Pikachu for almost the entire series.
As a matter of fact, Mari Iijima reprised her role as Lynn Minmei in ADV's 2006 English dub of the 1982 anime Super Dimension Fortress Macross. Unfortunately I can't find any clips of the dub, but here's an English interview with her about the role.
As amazing as that would be the answer is no. A majority of the voice actors in the industry would have to be fluent in English for that to happen, and that is sadly not the case here.
EDIT :A lot of Japanese voice acting tends to be 'good', but at the same time it's a lot different than the type of voice acting in English. With English, we kind of strive for natural sounding voices that sound normal. Whereas in contrast with Japanese voice acting, especially in Anime, nobody in real life talks like any anime character. While overacting is seen as a sign of a bad actor in the West, over in Japan it's kind of the opposite. And I feel because of this, it's kind of easier to come across as a good voice actor in Japanese than it is in English. That and directors are super picky about VAs. Some directors will go through hundreds of auditions for characters that aren't even the protagonist.
Also, would you like to watch an anime where all the jokes and puns which kind of make sense in Japanese, would be lost in translation?
As an answer said already the most prominent case of that happening is in the case of Pikachu.