It isn't linguistically true, at least. David Fried writes:
What’s with the movie convention of representing 19th century American speech as lacking contractions? I was just enjoying the new version of “True Grit” by the Coen brothers—in fact it’s been a long time since I had so much fun at a movie. As I figure it the action is set in 1878. Much of the pleasure of the movie is the oddly formal and elaborate diction of the characters, taken straight from the Charles Portis novel. I actually find a lot of it true to my conception of the period, if rather stylized, except for the absurdity of pronouncing all contracted auxiliaries in full. Ethan Coen was specifically asked about this in a Newsweek review, and replied rather ambiguously “We’ve been told that the language and all that formality is faithful to how people talked in the period.”
I haven't seen True Grit yet, and so I don't know whether it really "pronounces all contracted auxiliaries in full". In the Newsweek interview that David cites, there's no elaboration beyond the quoted sentence:
Q: Did people actually not talk with contractions at that time?
Ethan: We’ve been told that the language and all that formality is faithful to how people talked in the period.
Charles Portis originally published "True Grit" as a serial in the Saturday Evening Post in 1968, and in somewhat changed form as a book. The story is narrated in the first person by Mattie Ross as an old woman in 1928, describing her adventures many years earlier at the age of fourteen.
To start with, it's not the case that the 1968 Portis novel represents its characters as speaking entirely without contractions. One counterexample among many, from page 19:
He said, "Well I killed the wrong man and that is why I am here. If I had killed the man I meant to I don't believe I would have been convicted."
These two sentences have uncontracted "that is", "I had", and "would have", but contracted "don't". The treatment of other contractable verbs is also variable. Thus on p. 61
He said, "If I'm going up against Ned Pepper I will need a hundred dollars. I have figured out that much."
And on p. 180
"That's very well!" said Lucky Ned Pepper. "Do you advise me to kill her?"
For some other examples of this variable treatment, compare
Rooster said, "I bet it does. Set right still and it won't bleed so bad."
"No, you won't," said I. "This man will not let you have your way. He is your boss and you must do as he tells you."
She must have seen the dismay on my face for she added, "It will be all right. Grandma Turner will not mind. She is used to doubling up. She will not even know you are there, sweet."
"No sir, not me. Never. A man will not work for a woman, not unless he has clabber for brains."
Is this degree of variation (whatever it turns out to be in quantitative terms) an accurate representation of 19th-century American speech patterns? Or was Portis using an exaggerated (if variable) lack of contraction as a way to project the personality of Mattie Ross, an elderly woman whose strong notions of right and wrong may have included some prescriptive feelings about contractions? As Merriam-Webster's Dictionary of English Usage explains,
Won't was among the contracted and truncated forms that Joseph Addison attacked in The Spectator on 4 August 1711. It seems to have been under something of a cloud, as far as the right-thinkers were concerned, for more than a century afterward. This did not, of course, interfere with its employment, and it was common enough to enjoy the distinction of being damned in the same breath as ain't in an address delivered before the Newburyport (Mass.) Female High School in December 1846 [..] The speaker termed both "absolutely vulgar."
What about the current movie version? Does it really avoid contractions entirely, or limit them even more than Portis did? Would such contractionless speech really be an accurate reflection of the way Americans talked in Arkansas and Oklahoma in the 1870s?
I don't know what Portis intended, and I don't know what the Coen brothers did. But I know that that informal American speech in the 1870s was far from contractionless, and in fact I suspect that it had roughly the same proportion of contractions as it does today. Therefore, what Portis (and the Coens?) did was either false archaism or poetic truth — or both.
For some background on the development of contractions, I'll quote Barren Brainerd, "The contractions of not: A historical note", Journal of English Linguistics 22, 1989:
In this essay I have tried to outline the progress of contracted not from its first explicit appearance at the beginning of the seventeenth century in monosyllabic forms through its linguistically productive phase in the eighteenth to its general acceptance in the nineteenth. There is no concrete evidence for its existence before the beginning of the sixteenth century–at least among literate users of the language–and some evidence that it evolved at around that time from intermediate, non-contracted, and partially contracted portmanteau forms like cannot and the now obsolete wonnot, shannot, and donnot. From the dramatic evidence presented here it would appear that fully contracted not originated among speakers of nonstandard English, later to be appropriated by the educated classes. The salient feature in the above discussion is the extraordinary productivity of the form both among standard and nonstandard speakers.
What does "general acceptance in the nineteenth [century]" mean, specifically? Being a quantitative sort of person, I decided to try a little textual Breakfast Experiment™ (OK, it's dinner time, but anyhow.)
Mark Twain's Tom Sawyer (published, conveniently, in 1876) has 58 instances of won't, and just one of will not – in the author's preface:
Although my book is intended mainly for the entertainment of boys and girls, I hope it will not be shunned by men and women on that account, for part of my plan has been to try to pleasantly remind adults of what they once were themselves, and of how they felt and thought and talked, and what queer enterprises they sometimes engaged in.
There are 223 instances of don't, against just one instance of do not, which occurs in a context that is not altogether irrelevant to True Grit:
There is no school in all our land where the young ladies do not feel obliged to close their compositions with a sermon; and you will find that the sermon of the most frivolous and the least religious girl in the school is always the longest and the most relentlessly pious.
(Yes, I know that some instances of don't are third person singular, substituting for does not rather than do not. And I should really distinguish between dialogue and narration, as Henning Makholm observes in the comments. The current experiment is crude enough that I think it's reasonable to ignore such things for now.)
For these cases of not-contraction, it would hard for a modern novel to swing any more strongly in the pro-contraction direction. Looking at a couple of instances of is-contraction, we find that Tom Sawyer has 226 instances of it's, and 44 of it is (84% contraction); 197 instances of that's, as against 9 of that is (96% contraction). These are a bit more balanced, though still strongly weighted in favor of contraction.
How do these numbers stack up against a more modern novel? Picking one more or less at random, let's take a look at James Lee Burke's 2008 Swan Peak. Looking at not-contraction first, Burke's novel has 27 instances of won't versus 1 of will not; and about 319 instances of don't versus 7 instances of do not. Though the comparison to Tom Sawyer is not clearly distinct from chance variation, both changes are in the direction of a lower frequency of contraction, not a higher one.
In a similar sample of two cases of is-contraction, Swan Peak has about 255 instances of it's, and 26 of it is (91% contraction, as opposed to 84% in Tom Sawyer); and about 179 instances of that's, versus 9 of that is (95% contraction, as opposed to 96% in Tom Sawyer).
So to sum up, the 2008 novel has a lower empirical frequency of contraction in three out of the four cases checked. Obviously, a credible investigation would need to look at larger number of books from both periods, and should also check a larger number of contractions, and distinguish between dialogue and narration, and so on. But I take these numbers as zeroth-order support for the hypothesis that rates of contraction in vernacular literature (and presumably in everyday speech) have not increased over the past 140 years.
[When I say "about N instances", I'm extrapolating from the limited of 100 hits that Kindle gives. Thus in the case of Swan Peak's count for don't, the fact that Kindle found 100 hits in the first 2165 "locations", out of a total of 6898 "locations" in the book, means that there should be about 100*6898/2165 = 318.6 hits in the whole book. The only ways I've found to improve on this method are quite time-consuming and thus inappropriate for the Breakfast Experiment modality.]
I can't do exactly the same sort of check on True Grit, because there's no Kindle version, and the Google Books version is the "snippet view" mode that only reports the number of pages with hits. But using page-counts as a crude proxy for word-sequence counts, we find the following:
Comparing contraction percentages in the three works:
||Swan Peak||True Grit
So True Grit (the novel) has definitely got a lower frequency of contractions than the other two works, even though it's not in fact contraction-free; and this pattern is not a true picture of the 1870s southern or south-midland vernacular that its characters (like Mark Twain) presumably spoke.
If True Grit (the remake) has an even lower frequency of contractions, its picture of "how people talked in the period" is even less true, at least from the perspective of mere historical fact.
[Update -- Geoff Pullum reminds me that while "it's" and "that's" are indeed contractions, there are good arguments for treating "don't" and "won't" as irregular inflected forms. It remains true that a speaker or writer has a choice between (for example) "will not" and "won't", so the diachronic and sociolinguistic discussion is not crucially affected by this distinction (though perhaps it helps to explain some of the differences in the relative frequency of choices).]