“The system more closely resembles punishment than due process,” the lawsuit said.

Because people are not scheduled for specific times in court, they may have to take off an entire day from work, the suit says, and when they finally go before a judge, their appearance may last just a few minutes before the case is postponed to another date.

Defendants charged with misdemeanors in the Bronx regularly see their cases languish far past the 60- and 90-day speedy trial limits set down in state law for various low-level offenses, because delays caused by the courts’ crowded calendar rather than the district attorney are not counted toward the limit. Defendants who wish to go to trial must often wait years and sometimes never get their day in court, a 2013 study showed.

The delays are caused by a shortage of judges, court officers and court reporters that keep trial parts idle and locked, the lawsuit said. Misdemeanor defendants must wait on average 642 days for a bench trial and 827 days for a jury trial in the Bronx, far longer than in New York City’s other boroughs, according to the lawsuit. It noted that over 500 misdemeanor cases had been pending for more than two years.

In one case, a 40-year-old single mother was charged in 2012 with a count of misdemeanor assault; it was not until November 2015 — 1,166 days after her initial arraignment — that she received an adjournment in contemplation of dismissal, a disposition that requires no admission of guilt, according to the lawsuit.

Another man, Joseph Bermudez, a 36-year-old Bronx resident, was arrested in 2012 on suspicion of drunken driving, the lawsuit says. He received a bench trial in December 2015 — 1,255 days after his initial arraignment — and was acquitted, the suit says. In a phone interview, Mr. Bermudez, a warehouse manager at an appliance company, said he had to use overtime as well as vacation days and personal and sick leave to attend many of the court appearances.

The defendants named in the lawsuit are Gov. Andrew M. Cuomo; the state’s chief judge, Janet DiFiore; and Lawrence K. Marks, the court systems’ chief administrative judge. The private firms that are helping to represent the plaintiffs are Emery Celli Brinckerhoff & Abady and Morrison & Foerster.

Lucian Chalfen, a spokesman for Judge DiFiore, who took office in January, declined to comment on the specifics of the lawsuit, but acknowledged that chronic delays in Bronx courts were a troubling phenomenon, one the judge highlighted in her first policy speech.