Is a lengthy combination of words with hyphens like “the worst not-technically-in-a-recession year in American history” a new fashion of writing?

It's (usually) intended as a form of light sarcasm or irony.

James P is saying that technically, if you take the official definition of a recession, squint hard and don't look too closely, you can say with a straight face that 2012 is not a recession year; however, everyone knows that regardless of what the official figures may say, 2012 really is a recession year for all practical purposes.

More normal forms of expression (such as quote marks) tend not to convey the intended tone in which this is said. An equivalent verbal/visual equivalent may be "air quotes" (where a speaker raises both hands above shoulder level and flicks their fingers to indicate "quote marks") — i.e. while quote marks would not add the desired emphasis when written, by adding them when speaking a certain extra emphasis is added that is beyond what the words alone would easily convey. One can do this with suitable tone of voice, but it is more easily lost on the hearers.

I think at the heart of it, this question is really just about hyphenated compound words. Granted, the "word" here is much longer than is typically seen.

The Grammar.ccc.commnet.edu grammar site explains that there are three types of compound words:

the closed form, in which the words are melded together, such as firefly, secondhand, softball, childlike, crosstown, redhead, keyboard, makeup, notebook;

the hyphenated form, such as daughter-in-law, master-at-arms, over-the-counter, six-pack, six-year-old, mass-produced;

and the open form, such as post office, real estate, middle class, full moon, half sister, attorney general.

The Englishplus.com/grammar site website advises:

Hyphens connect the words of a compound modifier that comes before the word being modified. Hyphens are not used this way with compound parts ending in -ly or made up of proper nouns or proper adjectives.

Incorrect: He is a well respected man.

Correct: He is a well-respected man.
(A compound modifier before the noun.)

Incorrect: That man is well-respected.

Correct: That man is well respected.
(The modifier follows the noun, no hyphen.)

Incorrect: That was a badly-punctuated sentence.

Correct: That was a badly punctuated sentence.
(Modifier ends in -ly, no hyphen.)

Incorrect: The South-American rain forest is home to hundreds of species of hummingbirds.

Correct: The South American rain forest is home to hundreds of species of hummingbirds.
(Modifier is proper, no hyphen.)

You may also want to look at When should compound words be written as one word, with hyphens, or with spaces? as it has some good information on the subject.

The use of way-too-long-to-be-taken-seriously strings of hyphenated words is definitely informal. If it's gaining traction in "serious" newspapers like the NYT, that's most likely a recent phenomenon. I know it has been used for humor (rather than sarcasm or irony) for quite some time, at least since Douglas Adams in his Hitchhiker's Guide series. One example (from The Restaurant at the End of the Universe):

All this way, thought Zaphod, all this trouble, all this not-lying-on-the-beach-having-a-wonderful-time, and for what? A single chair, a single desk and a single dirty ashtray in an undecorated office.

I know his writing style influenced my own, and shortly after discovering his books, my writing took on many, many more hyphens. While I've since toned it down, I am not ashamed to admit that my fellow Adams-reading friends and I would often write letters to each other purposely trying to string together as many hyphens as we possibly could, obviously for our own amusement.

Extraordinarily-long-hyphenation is just a rather-modern-way of highlighting sarcasm and emphasizing continuity-of-idea-through-several-words, while also maintaining continuity-of-expression without sacrificing clarity or readability.

And no, it is not formal.

It's used. Rightly or wrongly. And stacked pre-modifiers without any hyphens can be worse than those using them. The following examples are found in grammar.about.com (Richard Nordquist):

Examples and Observations:

...

Ineffective

"The board also gave third reading to a Foothills Boulevard Landfill gas emission reduction credits transfer contract authorization bylaw." (from the Prince George Citizen [British Columbia], quoted by The New Yorker, June 27, 2011)

.........

Effective

"If you're unfamiliar with the joy of Ménière's (and I hope you are), imagine a floor-warping, ceiling-spinning, brain-churning, think-you're-gonna-die-and-afraid-you-might-not hangover and multiply that times the aftermath of a power outage at the all-you-can-eat Chinese buffet. That's Ménière's." (Kristin Chenoweth, A Little Bit Wicked: Life, Love, and Faith in Stages. Touchstone, 2009)

Multiple-hyphen compound adjectives have been in use for considerably longer than some people might think. For example, Marilla Ricker, I Don't Know, Do You? (1916) has this sentence:

The great Ingersoll had it right when he said it [the Calvin Baptist Church] was the damned-if-you-do-and-the-damned-if-you-don't church.

A non-adjectival instance involving similar hyphenation of almost the same words appears in Ernest Newman's review of "Mr. Le Gallienne's 'Religion of a Literary Man'" in the University Magazine and Free Review (March 1, 1894):

But even the philosophy of doing nothing has this uncomfortable side to it, that you do not know whether that will please God! You are in the pleasing old dilemma of You'll-be-damned-if-you-do, you'll-be-damned-if-you-don't.

If compound adjectives with three hyphens qualify as "lengthy combinations," we can find even earlier examples. For example, from "Preface to Our Second Decade" in Fraser's Magazine (January 1840):

In prose and verse, in speech and song, we were stunned by the profound affliction of the military Macbeaths, and the no-longer-gold-coated Peachums and Lockits of the imperial court.

From Henry Lytton Bulwer, "The Life of Lord Byron," in The Complete Works of Lord Byron (1837):

It was perhaps the not-to-be-satisfied satisfaction of a morbid mind, as well as the embarrassments of the irregular liaisons, and an ill-regulated fortune, which first induced him [Byron] to turn his thoughts upon marriage ; and there seems to have been something of seriousness in the admiration he entertained for Lady Elizabeth Forbes.

From Jeremy Bentham, Indications Respecting Lord Eldon (1825):

the consequence was—that, for every actual attendance, the Master, instead of 6s. 8d., received 1l., and that, even if inclined, no Solicitor durst omit taking out the three warrants instead of one, for fear of the not-to-be-hazarded displeasure, of that subordinate Judge and his superiors.

And from a letter of January 4, 1776, in Letters of the Late Ignatius Sancho (1782):

You would, like the fabled Pelican—feed your friends with your vitals—blessed Philanthropy! oh! the delights of making happy—the bliss of giving comfort to the afflicted—peace to the distressed mind—to prevent the request from the quivering lips of indigence!—but, great God!—the inexpressible delight—the not-to-be-described rapture in soothing, and convincing the tender virgin that "You alone," &c. &c. &c.

The second quotation in Edwin Ashworth's answer reminded me of an instance of multiple-hyphen compound adjectives from my youth—the hugely popular (in the United States) anti–Vietnam War song "I-Feel-Like-I'm-Fixin'-to-Die Rag" by Country Joe & the Fish, released in 1967.