A false trail: bouncing baby checks
The first instance of "bouncing checks" that an Elephind newspaper database search turns up uses bouncing not in the sense of "rubber-ball-like" but (evidently) in the sense of "healthy-baby-like." From "With Love—From Mother," in the Wheeling [West Virginia] Daily Intelligencer (February 26, 1900), reprinted from the Lewiston [Maine] Journal:
There are quirks and scratchy quavers of the pen/ Where it struggled in the fingers old and bent./ There are places that he has to read again/ And ponder on to find what mother meant./ There are letters on his table that inclose some bouncing checks;/ There are letters giving promise of profits on his "specs,"/ But he tosses all the litter by, forgets the golden rain,/ Until he reads what mother writes from up in Maine.
In this sentimental and not very metrical piece, the checks are bouncing with fresh-faced potential, not with the awful snap of insufficient funds.
The true path: bouncing rubber checks
Although (as noted in lbf's answer) Etymology Online dates the association of bounce (in the sense of return without payment) with check to 1927, newspaper database searches find references to it from as early as 1920—in particular, in connection with the term "rubber check." The earliest person to point out this association in print, as far as I can tell, is Damon Runyon. From "Damon Runyon on Slang" in the Washington [D.C.] Times (November 5, 1920):
A promissory note is nowadays called a "Kathleen Mavourneen." You remember the old song, 'It may be for years, and it may be forever?" A bad check is called a "rubber" check, because it bounces back when it hits the bank.
Two other relevant instances of "rubber check" from the first half of the 1920s show up in newspaper database searches as well. From "Reveal 'Subsidies' Paid by Ky. Jockey Club: 'Expendes' for Publicity Cut Big Hole in Receipts," in the [Chicago, Illinois] Collyer's Eye (December 22, 1922):
As soldiers in the ranks, with neither personal glory nor money our expectation, we will give our uttermost to revive the sport of kings in the second city in America. But our readers must not, and should not, expect us to sacrifice our self-respect and integrity by furthering the schemes of nickel-grabbers, dreamers, "Rubber check" purveyors[,] irresponsibles and mercenaries.
And from "Floyd Fitzsimmons' Troubles." in the New York Clipper (December 7, 1923):
Floyd Fitzsimmons, who promoted the fight arena at Michigan City, which is "just around the corner from Chicago," is reported in bad with the government over the admission tax on the Leonard-Kansas match. The gate for that affair was $54,000 and the Internal revenue department claims the 10 per cent was not paid. Fitzsimmons was in line to make plenty of dough from the Michigan City plant, but it was taken away from under his nose when the arena passed into the hands of a receiver. Jack Kearns was a close friend of Fitz and for him arranged the meeting between Jack Dempsey and Billy Miske at Benton Harbor. Fitzsimmons' money difficulties are reflected in several "rubber checks" alleged to have been given fighters appearing on his cards.
'Rubber checks' bouncing back in the period 1926–1927
The national vogue of "rubber check" and/or "bouncing" seems to have commenced in 1926–1927. Newspaper database searches yield seven relevant matches during the 17 months from late July 1926 to late December 1927. Here they are.
From "Speeder Found in San Quentin," in the Livermore (California] Journal (July 24, 1926):
A little thing like a speeding charge doesn't worry Mr. Furnish. He is enjoying a vacation of from 1 to 14 years in that well-known health resort, San Quentin, where he will be given plenty of time to ponder over the inadvisability of passing out "rubber" checks, the kind that bounce back.
From "Forged Check Passers Have System," in the [Healdsburg, California] Sotoyome Scimitar (December 24, 1926):
This is the open season for the accommodating merchant who blithely cashes checks for strangers. In street parlance there is a name for the merchant; He is called a boob and an easy mark. There is also a name for the checks: They are called "rubber checks," the term meaning checks that come back and slap the hand of the casher of them something like the "come back" of a rubber band which has been stretched taut and one end released.
From "The Hop Scotch," in the Stanford [California] Daily (January 5, 1927):
Other classy players whom Duncan will have to beat if he wins the aforementioned fat (but not bouncing) check, are Joe Kirkwood, the famous Australian trick-shot wizard; Mike Brady, who once got three holes in one in a single round; Johnny Farrel and "Wild Bill" Melliorn, two great players who are always threatening to win the United States open championship; Joe Turnesa, who almost did last year; George Von Elm, the United States amateur champion who beat Duncan badly in the last British open; and Willie Hunter, a former British amateur champion.
From Robert Small, "Girl Told Sheriff" in the San Bernardino [California] Sun (January 28, 1927):
A few days before the Bank of Buda [Texas] was rifled of so much of its spot cash the co-ed bandit la said to have issued a number of very lively "rubber" checks. Where she owed $150 she sent a check for $250. saying the excess was for "good measure." The recipient, of the checks felt a wonderful elation over the generosity of their girl debtor until the checks bounced back.
From "Make Plenty Money on 12-Cent Cotton," in the the Bartlett [Texas] Tribune and News (February 4, 1927), reprinted from the [Jourdanton, Texas] Atascosa News-Monitor:
Gee McGee tells how to make cotton pay at 12 cents a pound. We know your readers will be interested in finding out just how it can be done, so we give Gee's ten specific rules as follows:
"1. Tent a farm for part of the crop and shoot your landlord if he ever mentions his part.
"2. Buy your guano on credit. Steal your mule feed and plow tools from your neighbor. Give rubber checks for your groceries.
"3. Stay away from church so's your preacher won't expect you to pay him anything and if he sends a collector around, why turn him down.
From "Rainproof," in the Eagle [Colorado] Valley Enterprise (June 10, 1927), reprinted from Judge:
Spriggs—Strohm doesn't save a cent. What would he do in case of a rainy day?
O'Nell—Write a couple of rubber checks.
From "Speeder Has to Make Good on Bad Check," in the Livermore [California] Journal (December 21, 1927):
There are times and places when "rubber" checks may be cashed with full benefit to the casher but W. F. O'Keefe, Stockton business man, will agree that a Justice court and in payment of a fine are neither the correct place or time.
It appears that bad checks have been called "rubber checks" for at least long as they have been described as "bouncing." Writing in 1920, Damon Runyon offers a clear and concise explanation for the slang term: "A bad check is called a "rubber" check, because it bounces back when it hits the bank." Interestingly, most of the earliest references to 'bouncing" in connection with faulty checks adopt Runyon's wording: a bad check doesn't simply "bounce"; it "bounces back" to the person trying to redeem it for cash at a bank.
References to both "bouncing checks" (in a 1900 poem) and "rubber checks" (alluding not to bank drafts but to customer service tokens made of "worthless rubber," and dating to 1880 at least) apparently antedate by decades any use of the terms in the sense of "bad checks." But the meaning that the poster asks about had clearly emerged into use by 1920, in the form of "rubber checks" that "bounce back." The current sense of "bouncing a check" remains impressively true to the original conception of the slang term.