Thirty-two thousand federal workers have appealed their furloughs, and the 200 employees on the board are attempting to organize the cases and hold hearings, the Post said.
The board, which is charged with making sure employees are treated fairly, was not prepared for the volume of work, The Washington Post reported.
The increase has broken fax machines and crashed Internet servers, and the workload is six times what the agency is used to.
The office is consolidating some appeals that were issued by the same government official, the Post said, but the even the hearing rooms are too small to hold a large group of workers.
"If we're having consolidations with 50 people, how are they going to fit in these small rooms?" Jerry Cassidy, the chief of the Washington regional office, said.
If Congress does not stop the second round of cuts set for early next year, more cases could come in.
Fewer than five dozen cases have been heard so far, the Post said, and the judges have affirmed the furlough decisions.