It is hard to walk the halls of the Criminal Courts Building in Manhattan without encountering stories of crimes committed by people with mental illnesses. On Monday, two separate murder cases were being tried against men who had presented evidence they were psychotic when they stabbed their victims to death.

In a third courtroom, a homeless man with a history of mental illness, who had been jailed several times before for minor crimes, was scheduled to be arraigned on attempted murder charges after being accused of stabbing a street vendor in the chest with a pair of scissors.

Those are the nightmare cases, defense lawyers and prosecutors say, horrific situations in which people who never received effective treatment for mental illnesses ended up committing violent crimes.

But every day, dozens of people with fragile psyches are arraigned in Manhattan for minor crimes. Many cannot make bail and are remanded to Rikers Island, where they seldom receive the medical care they need, criminal justice experts said. These defendants often leave jail with a weaker hold on sanity than when they entered. A few go on to commit worse offenses.