“No Democrat has been elected president without carrying Texas” held true from 1845 until 1992 (when Bill Clinton was elected president), 1996 (when Bill Clinton was re-elected president) and 2008 (when Barack Obama was elected president). A similar presidential saying holds that no Republican can be elected president without winning Ohio.
 
Although this principle could have been applied much earlier in the twentieth century, the phrase seems to become popular since 1976 (when Jimmy Carter carried Texas and was elected president).
 
       
30 October 1970, Stevens Point (WI) Daily Journal, “Democrats’ Texas Troubles” by Rowland Evans and Robert Novak, pg. 4, col. 4:
The implications reverberate on the national level. A win by Bush would underline the difficulty of any Democrat carrying Texas against President Nixon in 1972, and a winning Democratic presidential candidate without Texas is hard to imagine.
(George Bush against eventual winner Lloyd Bentsen for U.S> Senate from Texas—ed.)
 
26 August 1972, Dallas (TX) Morning News, “McGovern Sees Texas Chance” by Sam Kinch, Jr., sec. A, pg. 19:
But it may turn out to have been a horrid misuse of the time available to a man who says he can’t win the Nov. 7 election without carrying Texas.
   
Google News Archive
24 April 1976, Free Lance-Star (Fredericksburg, VA), “The Texas establishment likely to support Carter” by Rowland Evans and Robert Novak, pg. 4, col. 4:
Gov. Briscoe, who voted for Wallace at the 1972 Miami Beach convention, says a Democrat cannot be elected President without carrying Texas and Carter is the one Democrat who can do it.
   
27 June 1976, Dallas (TX) Morning News, “Carter camp Texas leaders confident,” sec. A, pg. 15:
He (State Democratic Chairman Calvin Guest—ed.) said Texas would be a critical state for Carter and added that no Democrat in the 20th Century had been elected president without carrying Texas.
     
9 October 1980, New York (NY) Times, “Texas Looming as a Close Battle Between President and Reagan” by Hedrick Smith, pg. A1:
In this century, no Democrat has won the Presidency without carrying Texas.
       
Google News Archive
26 October 1980, The Ledger (Lakeland, FL), “The winner could be the loser” by Tom Raum (AP writer), pg. 15A, col. 5:
No Democrat has ever been elected president without carrying Texas—and this state is bound to be a major factor again this year.
   
Google Books
February 1984, Texas Monthly, pg. 126, col. 3:
Texas is essential for Mondale; no Democrat has ever been elected president without carrying it.
     
Google Books
The American elections of 1984
By Austin Ranney (American Enterprise Institute for Public Policy Research)
Durham, NC: Duke University Press
1986
Pg. 146:
One is Texas—no Democrat has ever been elected president without carrying the Lone Star state. The other is Ohio — no Republican has been elected president in this century without carrying Ohio.
     
New York (NY) Times
Bentsen Casts Two Ballots for Himself, Legally
By WARREN WEAVER Jr., Special to the New York Times
Published: Sunday, October 30, 1988
(...)
No Democratic candidate has ever been elected President without carrying Texas.
   
Google Books
Conspiracy of One:
The definitive book on the Kennedy assassination
By Jim Moore
Fort Worth, TX: Summit Group
1990
Pg. 9: 
No Democrat had ever been elected President without Texas’ electoral votes in his column.
     
Google Books
The Truth of the Matter:
My life in and out of politics
By Bert Lance with Bill Gilbert
New York, NY: Summit Books
1991
Pg. 74:
It is a truism of twentieth-century American politics that no Democrat can be elected President without carrying Texas. 
   
16 February 1991, Dallas (TX) Morning News, “Bush vs. Bentsen” by Sam Attlesey:
They note the trivia fact that no Democrat has been elected president without carrying Texas.
     
New York (NY) Times
THE 1992 CAMPAIGN: Political Pulse—Texas; In Texas, Three ‘Neighbors’ Battle for a Vital Prize
By R. W. APPLE Jr.,
Published: Sunday, October 4, 1992
(...)
For Mr. Clinton, with his strength elsewhere, the stakes are not quite so high. Still, top Democrats constantly remind one another that no member of their party has been elected President without carrying Texas since it entered the union in 1845.
   
The Independent (London)
The US Preseidential Elections: Perot fly in Clinton’s ointment
From JOHN LICHFIELD in San Antonio
Sunday, 11 October 1992
(...)
Mr Clinton can probably afford not to win here (though no Democrat in two centuries has become president without carrying Texas).
   
Abilene (TX) Reporter-News
McNeely: What if Dems win statewide in 2010?
Sunday, August 9, 2009
(...)
A Democratic presidential candidate hasn’t won Texas since 1976. There had been an adage that a Democrat couldn’t become president without carrying Texas — until Bill Clinton did so in 1992 and again in 1996.