Now representatives from each state are in talks with party leaders to come to a solution, in the hopes of avoiding a meltdown and a fractured party going into (and coming out of) Denver.
Gov. Howard Dean, chairman of the Democratic National Committee, said he thought it was "very unlikely" that Florida and Michigan delegations would be seated at the national convention without some concessions.
"I think it's very unlikely that Florida and Michigan, given how close this race is, are going to be seated as-is," he told Face The Nation host Bob Schieffer. "But everybody's going to work very hard to find a compromise within the rules that's fair to both campaigns that will allow Florida and Michigan in the end to be seated."
Dean stressed that, in light of talk of a "do-over" of the vote in those two states, the party will not foot the bill should it come to states holding a new primary election.
"The Democratic National Committee is not going to pay for it because, right now, our job is to tell the American people about Senator McCain's record on Iraq and the deficits and so forth, and convince the American people that our nominee is better than Senator McCain. And that's what we're going to be using our resources for."
Dean hoped that the issue could be resolved sooner rather than later, as the dispute has the potential of being damaging to the ultimate victor. "We don't think knock-down, drag-out fights are ever good between Democrats," he said. "We'd much prefer to have this all settled well before we get to the convention, if that can happen."
Also on the program, Sen. Bill Nelson, D-Fla., and Sen. John Kerry, D-Mass., talked about the importance of coming to a resolution to the delegate dispute in Flordia and Michigan, without resorting to a re-writing of the rules.
Nelson, a supporter of Hillary Clinton, said if the DNC will not seat Florida's delegates from the January 29 primary, "then the only thing I know to do is to do it over." He suggested a new election (possibly a mail-in vote) would cost $6 million, which the Florida Democratic Party would have to raise themselves.
Kerry, who has been campaigning for Obama, said the Illinois Senator would accept whatever deal is reached between the states and the party: "Let's go forward. Let's get this done in a way that allows Michigan and Florida to be at the convention, brings the party together, unites it - which is what his candidacy is about - and resolve this issue fairly."
Later in the program, Democratic strategist and CBS News consultant Joe Trippi and Republican strategist Howard Rollins talked about the debate over delegate sin Florida and Michigan and its potential to unite or shatter the party.
"I think you have to be very careful when you alienate voters, particularly in your base," Rollins said.
"You know, my sense is that the Republican side was like a NASCAR race in which the front-runners kind of get knocked back and forth, and, obviously, John McCain survived. The Democrat race is kind of like a Muhammad Ali-Joe Frazer fight - they're pounding each other, round by round. And it's pretty fascinating to watch.
"At the end of the day, I expect the party to be unified again, and John McCain to be a very strong candidate. And it will be a very close election."
Trippi was more cautious about the dangers a protracted primary battle would pose for the ultimate nominee, saying that when you have a big fight, the other team wins: "In 1980, when Senator Kennedy went after President Jimmy Carter, we went into that convention, the Democratic side, with a fight on our hands. And Reagan won [the general election].
"Democrats cannot go into this convention fighting. I think we've got to have this settled."
Read the full "Face the Nation" transcript here.