Neytiri in Avatar switches between Na'vi accented broken English and her native tongue.
That's not about intelligence - she learns more English through the film, after having only initially learned a small amount. Jake's skill at the Na'vi language is actually a better example in terms of the description.
The mutants in the obscure René Laloux film Gandahar (called Light Years in the English translation) said everything in both the past and future tense (example: "was will be" instead of "is").
However, they play with the trope; the mutants are trying to tell the main character that he needs to Time Travel...
Lampshaded in Murder by Death: Lionel Twain berates Inspector Wang (a parody of Charlie Chan) for his failure to "say his pronouns and articles." Fridge Brilliance applies when its revealed that Twain is actually Wang's foster father, meaning that Wang was raised in an English-speaking environment and is speaking that way on purpose. No wonder Twain is annoyed.
Averted (or subverted) with Tia Dalma from Pirates of the Caribbean. Besides having a thick accent, she speaks in a non-standard vernacular (such as "him carve out him heart, lock it away in da chest"). However, she's clearly a very intelligent, sly and mysterious character, and her unusual vernacular makes her even more attractive to some. Some fanfics like this one have her speaking in the third person as well.
She's not supposed to be speaking English, but rather Jamaican Patois, an English-based creole. It actually does sound a lot like this.
Examples from westerns deserve a whole section, as Native Americans mostly speak in short sentences, dropping articles and stuff. Sometimes that's also a case of Eloquent In My Native Tongue.
To begin with, Nevada Smith (1966). Wounded hero is healed by a bucolic tribe of natives. When he comes to his senses, he's greeted like this: "You come back to us in trouble. And in pain. You are welcome." — "How long?" — "Many days. You talk in fever." And so on.
Parodied in Maverick: The Native Americans can speak English perfectly well, but the Russian Duke wants a more "authentic" Western experience.
From Alexander, the Persians (and much of the Greeks/Macedonians) speak fairly and eloquently ("If only you were not a pale reflection of my mother's heart") whilst the Baktrian Roxane speaks in this manner: "Great man, Alexander? You I kill now." Potentially justified in showcasing that Roxane was not very fluent in the native language of the Greeks, while her father Oxyartes speaks perfectly fluent.
Subverted in, of all things, an outtake shown during the ending credits of Rush Hour when Jackie Chan points out that Chris Tucker cannot speak even three words of Chinese.
Though perhaps ends up proving a real-life example when Jackie berates him by saying "Now you see how difficult I am!"
In the 1975 film The Wind and the Lion, the concept is parodied by the use of an old vaudeville joke. U.S. Secretary of State John Hay, seated at a diplomatic dinner next to a Japanese representative, asks him as each course is served, "Likee fishee?" and "Likee soupee?" (the "pidgin" English of the period); at the dinner's end, the Japanese gentleman rises and delivers a long and eloquent toast in English to President Theodore Roosevelt, and then, seating himself by Hay, turns to him and asks, "Likee speechee?" The same bit is used in one of the Charlie Chan films.
Played for laughs in Fear Of A Black Hat. A video is shown with an attractive, Asian singer with an impressively booming voice, but when she is interviewed, she can barely speak a word of English. The real singer, an overweight, unattractive woman, then confronts her. This was a parody of the video for "Gonna Make You Sweat (Everybody Dance Now)", which also featured a beautiful woman who it later came out did not do the actual singing. You can compare the original to the parody yourself.
In 7 Faces of Dr. Lao, the titular wizard casually switches back and forth between this and speaking perfect English.
Vanko from Iron Man 2 feigns this with Hammer, basically because he doesn't like him. Also, it serves as Obfuscating Stupidity, leading Hammer and his guards to underestimate Vanko.
Mimi-Siku from Jungle 2 Jungle speaks like this, which is odd since his mother (who raised him alongside a primitive South American tribe) is American and speaks perfect English. Mimi's speech usually leaves out all conjugations of the verb "be," but it's also rather inconsistent depending on the scene, as he uses either "me" or "I" where "I" would be appropriate.
"Me happy to be with you, Baboon." "I want to see Statue of Liberty, I go."
The Room: "Everybody betray me, I fed up with this world!"
Played for Drama in Threads, as the children born after the nuclear bombs drop can only speak in halting broken English because no resources can be spared to educate them note that is, barring sticking them in front of a TV set playing a ropey video recording of BBC schools programme, Words and Pictures.