The phrase a bit tight under the arms has been jocularly applied to a pair of trousers much too large for the wearer.

The earliest occurrence that I have found is from The Belper News (Belper, Derbyshire, England) of Friday 13^th March 1908:

A BEAUTIFUL FIT.

A second-hand clothes dealer was selling a suit to a very meek and easily satisfied customer. All went well until the breeks were tried on. Then—“Ain’t they a bit on the big side, mister?” asked the customer, timidly.
“Big? No fear. Beautiful fit! Let me brace ’em up. There! Now they’re lovely; and comfortable too, I’ll lay. Ain’t they?”
The customer wriggled.
“Not bad, mister,” he said, meekly; “but a little bit tight under the arms.”

Several British and Northern-Irish newspapers reprinted this story in 1908 and 1909.

In the USA and in Canada:—The same story appeared in the San Francisco Chronicle (San Francisco, California) of Monday 13^th April 1908, and was often reprinted, for example in the Newark Evening Star and Newark Advertiser (Newark, New Jersey) of Monday 4^th September 1911, in The Ottawa Evening Citizen (Ottawa, Ontario) of Monday 23^rd July 1923, and in The Albany Capital (Albany, Missouri) of Thursday 21^st November 1935.

A variant of the phrase occurs in “Mac” Spins a Few War-time Yarns, published in the West Middlesex Gazette (London, England) of Saturday 12^th November 1927:

Meeting a recruit coming away from the Q.M. store in all the glory of his new uniform, the officer asked, “Everything all right?” “Yessir,” replied the “rookie.” “Uniform fit you?” “Oh, yessir, all ’cept the trousis; they’re a bit tight under the arm-pits!”

The phrase has been applied to other things than trousers. For example to small cars—as in the following from the column Smiles, published in The Coshocton Tribune (Coshocton, Ohio) of Friday 2^nd November 1928:

Salesman—Then, will this car suit you?
Prospect—No, I’d like a size larger. This is a bit tight under the arms.—Paris Pele Mele.

Likewise, the following is from the column Day by Day, published in the Evening Telegraph (Dundee, Angus, Scotland) of Monday 7^th October 1929:

A famous motorist is having a car built for him. We can see him with the manufacturer testing his new machine and saying, “Let’s see; it’s a bit tight under the arms. I don’t like the bonnet you have attached to it. But I think it is all right behind.”

And this is from an article by Duncan Gillespie about the Scottish Motor Exhibition in Glasgow, published in The Windsor Daily Star (Windsor, Ontario) of Tuesday 3^rd December 1957:

At the small end was the tiny B.M.W. Izetta runabout at under £400—one spectator thought it looked very neat but might be a bit tight under the arms!

The phrase has also been applied to socks much too large for the wearer—as in the following from The Leader-Post (Regina, Saskatchewan) of Wednesday 17^th September 1941:

Just as we must joke about the new bride’s cooking, so knitters come in for their share of “teasing” about the socks they knit and send overseas. Some P.S.’s which recently have been affixed to letters home (notably to new knitters) are: “Socks were swell but a bit tight under the arms,” and “Thanks a lot for the socks, were grand even though I did have to take two steps to catch up to them.”

And this is from the column Try and Stop Me, by Bennett Cerf ¹, published in several U.S. newspapers in June and July 1964—for example in the Evening Herald (Shenandoah, Pennsylvania) of Thursday 4^th June:

Ed Condon ² writes jubilantly that his wife has at last completed knitting a pair of socks for him originally intended for a 1961 Christmas present. “They’re magnificent,” adds Condon, “but just the least bit tight under the arms.”

¹ Bennett Cerf (1898-1971) was a U.S. publisher and humorist.
² This may refer to the U.S. nuclear physicist Edward U. Condon (1902-1974).

A punning use of the phrase occurs in the following, from The Yorkshire Evening Post (Leeds, Yorkshire, England) of Wednesday 26^th April 1916:

Miss Ethel Levey ³ is being “fitted with a new song.” At present, “London Opinion” comments, it is just a wee bit tight under the arm.

³ Ethel Levey (née Grace Ethelia Fowler – 1880-1955) was a U.S. actress, dancer and singer.

This cartoon was published in several British and Northern-Irish newspapers in December 1909—for example in The Diss Express (Diss, Norfolk, England) of Friday the 3^rd:

MADE TO MEASURE.

“’Ow do they fit, Bill? Bit tight under the arms, ain’t they?”

And this cartoon was published in the Daily Mirror (London, England) of Monday 9^th December 1968:

“A bit tight under the arms for Tchaikowsky, isn’t it?”