of this year's competitive races raises the possibility that the 100-member chamber could be evenly split next year with 50 members caucusing with each of the two parties. That divide could have an immediate and potentially disruptive impact on the incoming administration of either Hillary Clinton or Donald Trump, especially when it comes to nominations, which require Senate confirmation.
"I think it would be partisanship on nitroglycerin," said Ron Bonjean, a veteran Republican strategist and former spokesman for both Senate and House leaders.
It's a rare situation for the 228-year-old legislative body and, while the Senate historian's office recounts "The Great Senate Deadlock of 1881"
and another smaller impasse in 1953, arguably the real precedent for figuring it out is relatively recent history that happened in the wake of the most extraordinary presidential election of modern times.
While the country was on edge in late 2000 watching the Florida recount and then the front steps of the Supreme Court to hear the Bush v. Gore decision in mid-December, newly-elected and returning members of the Senate were slowly coming to grips with another issue. Despite Al Gore's failed bid for the White House, Democrats actually gained five seats to even up the balance of power in the Senate.
With Dick Cheney coming in as vice president in 2001, the 50 Senate Republicans had the rightful claim to the majority with Cheney as a tie-breaking vote. But some Democrats, reeling from Gore's defeat and emboldened by their own gains -- after six years in the minority -- questioned why committee make-up should be anything other than an even divide.
One of the first items of business for a new Senate is a plan for committee assignments and budgeting. It's the backbone for how the Senate does its business. Committee breakdowns have historically mirrored the make-up of the overall chamber. For example, if the Senate breakdown is 60 members in one party and 40 in the other, then a 20-member committee could be split by the same proportion of 12 to eight.
The agreement for equal shares of committee seats -- Republicans held all the chairmanships -- was reached largely through a direct conversation between the two Senate leaders at the time, Mississippi Republican Trent Lott and South Dakota Democrat Tom Daschle.
"Tom and I had a very good relationship. It was one based on communication and chemistry," Lott said. "I have to tell you that working out that 50-50 agreement with Tom was the toughest job I ever had to do as majority leader."
"We attempted to allocate resources, committee membership, and legislative calendar matters on a 50-50 basis," Daschle said. "It required virtually daily conversation and cooperation between the two leaders. We had initial opposition from members in both caucuses."
Some Republicans recoiled at the idea that Democrats would have the same number of seats on committees given that Cheney was the incoming vice president, but ultimately they agreed to the plan, which made Daschle the majority leader for 17 days in January 2001 and then turned things over to Lott once Cheney replaced Gore as vice president.
"Compromise was not such a dirty word back then," said Bonjean, who was Lott's spokesman at the time.
The outlook for a repeat scenario just 16 years later is mixed.
"The optimistic view is the Senate now has a precedent for what to do in that circumstance. The pessimistic view would be, as partisan as things were then, they're more partisan now," said Mark Patterson, who was Daschle's policy director during the 2001 session. "It's quite conceivable that no agreement would be reached and the Senate would be thrown into some state of gridlock or chaos."
"There were a lot of raw feelings [in 2001]," said Rodell Mollineau, who served as a spokesman for Daschle in 2001 and later worked for Nevada Senator Harry Reid. "I don't necessarily think you're going to have that this time."
The biggest complication for the incoming president could be getting swift confirmations for nominees and other appointments. If committees aren't up and running, then there's no way to hold hearings or send a nomination to the floor for approval. That means Hillary Clinton or Donald Trump could spend weeks or even months without confirmed cabinet members.
As the Senate historian's office notes, the 1881 standoff lasted for what they describe as "11 chaotic weeks."
A resolution could come down once again to the relationship between the two Senate leaders. Kentucky's Mitch McConnell is expected to continue to serve as his party's leader and New York's Chuck Schumer is expected to take over as Democratic leader for the retiring Reid.
"I think that McConnell and Schumer might come closer to reaching a solution than Reid and McConnell would," said Lott. "They haven't screwed each other yet."
Today's climate also has some different dynamics than what faced Daschle and Lott in 2001 with the rapid rise of new media platforms that have become central to shaping political fights.
"We weren't beaten up with this Facebook and social media chatter that you have now," Lott said.
Of course, the 2001 power-sharing agreement was upended just months later when longtime Republican Sen. Jim Jeffords of Vermont left the party to become an independent and caucus with the Democrats, swinging control to them with a one-seat majority in the chamber as well as on all committees except the one dealing with ethics issues.
At that time, there were a handful of moderates in both parties who were casually and aggressively courted by party leaders to flip allegiances, but there are even fewer members in the ideological middle just 16 years later.