He was injured when he hit the wall in center after making the catch.
"There's a fracture there," Nationals Assistant General Manager Mike Rizzo said.
Rizzo didn't want to speculate on how long Bernadina might be out.
"We'll know [Sunday]. We'll let you know," Rizzo said.
Bernadina was recalled from Triple-A Syracuse on Wednesday, one day after Lastings Milledge was sent down to the Chiefs.
Bernadina is batting .250 in three games.
Washington will recall OF Justin Maxwell from Syracuse on Sunday. Maxwell is batting .226 with two home runs and four RBIs. Maxwell will not regularly start, Rizzo said.
Rizzo didn't want to quickly recall Milledge.
"He needs to get his at-bats down there every day," Rizzo said. "I want Milledge to play every day."
Bernadina was playing because Elijah Dukes was held out of the starting lineup for arriving late for pregame stretching.
"He was late for work," Rizzo said. "It's not fair for his teammates if he's late for work."
The Nationals also purchased the contract of infielder Alex Cintron from Triple-A Syracuse before the game to fill the need for a backup infielder. Shortstop Cristian Guzman was placed on the 15-day disabled list after Friday night's game because of a strained left hamstring in Monday's home opener while leaving the batter's box.
Guzman was hitting a team-high .515 (17-for-33) with four RBIs as Washington's leadoff batter.