“No question, 2010 redistricting hurt us,” said Kelly Ward, who heads the Democratic Congressional Campaign Committee. Asked whether the party could win the 218 seats necessary to take back the majority, she said: “There’s always a chance, of course.”

Rob Simms, Ward’s counterpart at the National Republican Congressional Committee, has a different take: “I think it’s very difficult” for Democrats to take back the majority under the current maps, he said.

The district lines drawn by Republicans after the 2010 census left Democrats with few real opportunities. David Wasserman of the Cook Political Report says just 28 House seats are truly competitive — and seven of those seats are held by Democrats. The party would need to win all of those, plus nine more solidly Republican seats, to win a majority.

The court cases in question today could change the balance of power in only a small handful of seats. Mark Gersh, a leading Democratic Party redistricting expert, estimated that his side could pick up two to three seats each in Virginia and Florida, and one to two seats in North Carolina.