“A bicycle can’t stand alone; it is two-tired” (a pun on “too tired”) is a frequently told bicycle pun with a long history. “Unless the wheel is two tired it can’t be ridden; and if the rider is too tired he can’t ride” was cited in April 1896; “Although the bicycle is never fatigued it does not run well until it is two-tired” was cited in August 1896.  The “two tired” bicycle pun (in its various forms) is one of the oldest of puns told in the United States.
   
 
12 April 1896, The Daily Herald (Grand Forks, ND), “Nubs of News,” pg. 4, col. 4:
The Park River Gazette has discovered something curious in the bicycle line: unless the wheel is two tired it can’t be ridden; and if the rider is too tired he can’t ride.
 
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14 August 1896, The L.A.W. Bulletin and Good Roads, pg. 229, col. 1:
ALTHOUGH the bicycle is never fatigued it does not run well until it is two-tired.
 
2 June 1898, Colored Citizen (Topeka, KS), “Tipped by the Way,” pg. 3, col. 4:
Strange, isn’t it, that a bicycle can’t go except when it is two tired?
   
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March 1922, The Pedagogical Seminary, “Notes on the Psychology of Recreation” by G. Stanley Hall, pg. 81:
A bicycle cannot stand alone because it is too tired (two tired).
 
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August 1952, Boys’ Life “Think and Grin,” pg. 50, col. 3:
Nit: Why can’t a bike walk?
Wit: It’s two tired. — Wayne Pierce, Columbus, Ind.
 
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February 1960, Boys’ Life, “Think and Grin,” pg. 90, col. 1:
Why can’t a bicycle stand up by itself? Because it’s two-tired. — Richard Williams, Brooklyn 6, N.Y.
 
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November 1962, Boys’ Life, “Think and Grin,” pg. 82, col. 2:
Joe: Why doesn’t a bicycle stand up by itself?
Moe: I don’t know. Why?
Joe: Because it’s two tired. — Carl Lundeen, Jr., Minneapolis, Minn
   
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The Rise and Fall of American Humor
By Jesse Bier
New York, NY: Holt, Rinehart and Winston
1968
Pg. 18:
The very lowest form of sound-effect punning is that in which only one of two terms has any truth at all: A bicycle cannot stand up because it is . . . two-tired.
 
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April 1970, Boys’ Life, “Think & Grin,” pg. 82, col. 1:
Quiz: Why can’t a bike stand up by itself?
Whiz: Because it’s two-tired. — Rosemary Monagle, Medford, Mass.
   
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Biggest Riddle Book In The World, Part 2
By Joseph Rosenbloom
Illustrations by Joyce Behr
New York, NY: Sterling Publishing Company, Inc.
1976
Pg. 102:
Why can’t a bicycle stand up by itself?
Because it is two-tired (too tired).
   
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Kids’ Funniest Jokes
Edited by Sheila Anne Barry
Illustrated by Jeff Sinclair
New York, NY: Sterling Publishing COmpany, Inc.
1993
Pg. 16:
Why can’t a bicycle stand?
Because it’s two-tired.
MICHAEL KILPATRICK, 10
   
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Tyrannosaurus Lex:
The Marvelous Book of Palindromes, Anagrams, and Other Delightful and Outrageous Wordplay
By Rod L. Evans, Ph.D.
New York, NY: Perigee Book
2012
Pg. ?:
A bicycle can’t stand alone; it is two tired.
   
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Your Sense of Humor:
Don’t Leave Home Without It
By K B Chandra Raj
Victoria, BC: Trafford Publishing
2012
Pg. 114:
“Why doesn’t a bike stand up by itself? Because it’s two-tired.”