(CNN) - A bill allowing Kentucky GOP Sen. Rand Paul to simultaneously run for the presidency and re-election to his U.S. Senate seat in 2016 died earlier this week when the Kentucky legislature adjourned for the year.
The bill had passed the Republican-controlled state Senate, but stalled in the Democratic-controlled state House of Representatives.
"In Kentucky, you ought to run for one office at a time," Brian Wilkerson, a spokesman for Kentucky House Speaker Greg Stumbo, told CNN on Thursday. "The speaker's thoughts haven't changed on that."
The state's Democratic governor, Steve Beshear, is highly unlikely to call the legislature into special session to consider the measure. And that means if Paul's allies in the state legislature want to try again, they're going to have to wait until the legislature reconvenes next January.
By that time, a number of 2016 White House contenders may already be officially in the race.
GOP State Senate Majority Leader Damon Thayer, a supporter of Paul's, has argued that Kentucky's law prohibiting running for two offices at the same time is meant to apply only to state-level offices – not federal offices such as the presidency, the U.S. Senate or the U.S. House.
"Senator Paul is very popular here, and I personally don't want to deny Kentuckians the chance to vote for or against one of their own for the Senate if he's also nominated for the presidency," Thayer told CNN in March.
A number of other states let politicians run for the House or Senate while also running for the White House, Thayer noted.
"It's not fair to Senator Paul if he can't do the same," he said. "Why should Kentucky be any different when it comes to one of their own running for president?"
But Florida GOP Sen. Marco Rubio - another potential 2016 presidential contender - argued earlier this month that candidates shouldn't run for the presidency while trying to hold onto another office.
"I think by and large, when you choose to do something as big as that, you've really got to be focused on that and not have an exit strategy," Rubio said during an April 2 appearance on the Hugh Hewitt radio show.
As with Kentucky, Florida law prohibits candidates from being on the ballot for two different offices. Rubio described that rule as "the right law."
Paul, a tea party favorite, is currently considered a likely contender for the 2016 Republican presidential nomination. If that bid doesn't pan out, a Senate re-election victory would allow him to maintain his foothold on Capitol Hill for another six years, starting in January 2017.
Several members of recent presidential tickets have essentially hedged their political bets by running for re-election while simultaneously pursuing higher office.
Wisconsin Rep. Paul Ryan ran for re-election to his seat in 2012 while serving as the GOP vice presidential nominee on Mitt Romney's ticket. Then-Sen. Joe Biden also ran for re-election in Delaware while simultaneously serving as then-Sen. Barack Obama's running mate in 2008.
In 2000, Joe Lieberman was re-elected to his Senate seat in Connecticut while losing his bid for the vice presidency when he and Vice President Al Gore, the Democratic presidential nominee, narrowly lost the election to George W. Bush and Dick Cheney.
–CNN's Steve Brusk contributed to this story