“We are in fact drowning in intelligence,” said Alain Bauer, a professor of criminology at the National Conservatory of Arts and Crafts in Paris.

He and others said there were structural problems, including the fact that France’s so-called S List, a database of people believed to have been radicalized, has over 10,000 names and is not ranked according to threat level.

Though most on the list never commit violence, others have now been responsible for gruesome headlines. Eight of the 10 men who staged the deadliest European terrorist attack in over a decade — the Paris killings on Nov. 13 — were on the S List and several had spent time behind bars, yet were able to sneak back into France and Belgium from Syria. Another suspect on the list, Amedy Coulibaly, had also been imprisoned on a terrorism conviction. Eight months after his electronic bracelet was removed by the French authorities, he killed a police officer and opened fire in a kosher supermarket in Paris in January 2015, leaving four more people dead in the Islamic State’s name.

“If you take your daily agenda, and you were to note down the birthday of every single person you know, it would be unmanageable” to try to wish them all a happy birthday, Mr. Bauer said. “You need to make a selection. We don’t know how to do that with the profiles of these people.”

Those kinds of suspects have created an awkward middle ground for the French authorities, and after a series of plots or attacks linked to the Islamic State over the past two years, there is more urgency to find new legal tools to deal with the problem.

After Mr. Abballa killed the couple in Magnanville, France, last week, a deputy in Parliament, Éric Ciotti, introduced a bill creating the status of “administrative detention” for those representing a security threat.

In effect, he was calling for rapid prioritization of the S List, and he said the bill would be aimed at immediately detaining hundreds of those deemed to pose the highest risk, placing them under house arrest or in a detention center.