A “ten gallon hat” cannot hold ten gallons of water. The “gallon” term is said to come from the Spanish word galon, a decorative braid worn on the hat. “Gallon hats” (yes, there were “two gallon” hats) date from about 1903, while “ten gallon hat” dates from about 1920.
   
 
Stetson Hats: FAQ
How much water does a 10-gallon hat hold?
3 quarts
   
Wikipedia: Ten-gallon hat
A ten-gallon hat is a type of cowboy hat. The origins of the name are unknown but there are two common stories, both centering around linguistic misunderstandings between the Texan cowboys and the Mexican vaqueros.
 
The first story tells that the vaqueros wore hats decorated with galloon (narrow braided trimming) around the crown. The Spanish word for “galloon” is “galón.” Texas cowboys misunderstood the word “galón” for “gallon,” and a false legend was born.
 
Another widely accepted etymology is that vaqueros would describe the hats as “tan galán” or “so gallant” and American cowboys misheard it as “ten gallon.”
 
A ten-gallon hat in fact holds less than a gallon of water, and a hat that held ten gallons would be un-wearable.
 
One popular manufacturer of this type of hat is the Stetson company. 
 
(Oxford English Dictionary)
ten-gallon hat, sombrero, a high-crowned, wide-brimmed hat of a kind esp. worn in the south-western U.S. (cf. STETSON);
1928 Daily Express 7 Oct. 3/7 She instinctively recognized that he was a cowboy, even though he did not wear a ten-gallon hat and a jacket embroidered with Mexican dollars.
1929 T. WOLFE Look homeward, Angel (1930) xxvii. 374 He removed from his head the ten-gallon grey sombrero.
1939 Amer. Speech XIV. 201/1 In the nomenclature of the South-western cowboy, sombrero is used interchangeably for hat, but the qualifying phrase of ‘ten gallon’ has been arrived at by a mistaken translation of a Spanish word. The word ‘gallon’..served to describe the braid with which a vaquero’s hat was trimmed..it should have been ‘galloon’.
       
24 December 1903, Trenton (NJ) Times, pg. 1, col. 2: 
City Clerk Harry B. Salter wants a pretty diamond pin to wear with his two gallon hat and cane.
     
12 October 1904, Trenton (NJ) Times, pg. 4, col. 3:
For campaign purposes an automobile is about as useless as a two-gallon hat, gold-headed cane and kid gloves.
 
19 January 1906, Washington Post, second part, pg. 1, col. 7:
“Oh, yes, of course, I am going to put on a three-gallon hat and gothic spats and run the lawn mower over my whiskers.
 
18 December 1912, Sandusky (OH) Star-Journal, pg. 4, col. 2:
A man may wear a gallon hat to church on Sunday, but still take a widow’s home away from her on Monday.
   
25 December 1913, Ogden (Utah) Standard, pg. 8, col. 4:
The shiny silk cylindrical hat, sometimes called “topper,” “high hat,” “plug,” “high dicer,” “stovepipe,” or “four gallon hat” by the irreverent, seems fated to disappear, after little more than one century of existence. In England, the place of its origin, it was long esteemed the every highest badge of respectability,...
 
(North American Women’s Letter and Diaries database)
Morrison, Anna Daly. “Diary of Anna Daly Morrison, July, 1920”
[Page 62]
midnight one Saturday night in a saloon, the truckers were boisterously discussing the topic nearest their hearts—Diamond T’s, the loads they can haul, their speed, gas consumption, etc.—when a cow puncher with a bandanna around his neck, and wearing chaps, high boots, spurs and a ten-gallon hat, suddenly elevated himself to a standing position on the bar and in a mighty voice shouted, “Listen, all you blankety-blank Diamond T truckers, this is to advise that we’re going to punch cattle here next Saturday night and to h—with Diamond T’s.” After the truckers leave the hotel
 
Morrison, Anna Daly, 1884-1957, Diary of Anna Daly Morrison, July, 1920, in Diary of Anna Daly Morrison, Those Were The Days. Boise, ID: Em-Kayan Press, 1951, pp. 446. [Bibliographic Details] [7-17-1920] S1105-D028 Morris:D1105-28
 
21 November 1921, New York

, pg. 3:
...American cow punchers in two-gallon hats…
   
18 March 1924, Galveston (TX) Daily News, pg. 1, col. 3:
Tall, with dark eyes, a face filled with latent humor, sinewy and raw-boned, topped with a ten-gallon hat, clothed in a gorgeous and resplendent shirt, worn chaps, high-heeled and appliqued boots, a slow drawl, an awful lot of courteous ways—he has it all.
 
27 May 1924, Evening State Journal and Lincoln Daily News (Lincoln, NE), pg. 1, col. 2:
“TEN GALLON” HAT
FOR THE PRESIDENT
When the crack polo team of the University of Arizona, at Tucson, on its way to New York to play, called on President Coolidge in Washington, the members presented Mr. Coolidge with one of the southwest’s famous “ten gallon” hats, which Senator Ralph H. Cameron, of Arizona, who accompanied them, is shown wearing.
     
25 September 1975, Charleston (WV) Gazette, pg. 33, col. 2:
“Gallon” as used in hat comes from the Spanish “galon” and a “galon” is a fancy braid of gold or silver which Mexican vaqueros wear around the brims of their sombreros. Five rows of decorative braid and the wearer had a “five galon” hat. Ten rows—“ten galons.” A “ten galon” hat was a pretty fancy hat although you could get a “15 galon” hat if you wanted one. But among cowboys the “10 galon” sombrero came to stand for a very good hat, while “galon” got corrupted to “gallon.”