Earl Truvia and Gregory Bright were convicted of a 1975 murder in New Orleans and sentenced to life. Their conviction was based on the testimony of a single eyewitness, a schizophrenic heroin addict who testified under a false name to hide her criminal history. Local prosecutors did not turn over this and other key evidence to the defendants. In 2002, both men were exonerated largely for this reason. After their release, they sued the office of the longtime New Orleans district attorney, Harry Connick, for letting his prosecutors railroad them into prison.

Under a landmark Supreme Court decision, Brady v. Maryland, prosecutors must disclose any material evidence that could exonerate a defendant. But because individual prosecutors are immune from being sued, the only way to hold the government accountable is if a court finds a systemic failure to train prosecutors properly on the Brady rule.

This would seem to be an easy bar to clear in New Orleans, where, as Mr. Truvia and Mr. Bright argue, Mr. Connick effectively had a policy of not turning over exculpatory evidence. He consistently neglected to provide any such training to his staff, even though the office’s failure to disclose exculpatory evidence led to the exoneration of at least 12 people since 1990. A former assistant prosecutor, they say, described the office’s unwritten policy as “when in doubt, don’t give it up.”

Yet, the United States Court of Appeals for the Fifth Circuit threw out the lawsuit, saying the men had not proved that Mr. Connick’s office had any policy to withhold evidence or that he had failed to train his prosecutors. Nor, it said, had they proved there were any Brady violations before their convictions.

The behavior of Mr. Connick’s office was also at issue in the 2011 Supreme Court case. John Thompson, convicted of murder and armed robbery, spent 18 years behind bars, 14 of those on death row, before a private investigator discovered that several prosecutors in Mr. Connick’s office had lied for years about a crime-lab report that ultimately led to Mr. Thompson’s exoneration.