Visiting the night small-claims court on Sutphin Boulevard in Jamaica offers a catalog of the irritations of city life, writ small. A livery-car driver was irate. Two former friends seemed about to spit. And Tashmina Muslima was dejected when her case against her former landlord was postponed until November, the next time an official Bengali interpreter would be available.

“I’m very surprised,” Ms. Muslima said outside Judge Purificacion’s courtroom, “because this is a very important place to come for justice.”

With court budgets cut in some 30 states, small-claims courts have been among the least noted casualties.