By RUKMINI CALLIMACHI JUNE 21, 2016

MAGNANVILLE, France — The first time Larossi Abballa appeared on the radar of French terrorism investigators, the only act of violence they could pin on him was killing bunnies.

He had joined a small group of men, all bent on waging jihad, on a trip to a snowy forest in northern France five years ago, when he was 19. There, they videotaped themselves slaughtering the rabbits, bought so the men could grow used to the feel of killing.

When he and seven others were later arrested, the authorities found that several of the men had saved the video of the slaughter on their cellphones, alongside footage of soldiers being beheaded, according to French court records. Mr. Abballa was eventually convicted on a terrorism charge and spent more than two years in prison.

In hindsight, it is not hard to see how that first act of brutality foreshadowed what happened last week: Armed with a knife, Mr. Abballa attacked a couple in northern France in the name of the Islamic State and left them to bleed to death.

Across Europe and the United States, law enforcement officials are struggling to reckon with attackers like Mr. Abballa and Omar Mateen, whose shooting rampage this month at a gay nightclub in Orlando, Fla., left 49 dead. They are men who clearly seemed to be building toward violent acts, and whose names had surfaced in terrorism investigations, but who avoided crossing legal lines that could tip off the authorities until it was too late.

With thousands of terrorism surveillance cases running at any given time, the European authorities say they are swamped and are in the difficult position of trying to head off attacks of which the only forewarning is often in the form of what someone thinks or what they are overheard saying. …

As the most junior member of the group, Mr. Abballa was not chosen to go, and that frustrated him. “I’m thirsty for blood, Allah is my witness,” he wrote in an email intercepted by the authorities. In another, he begged, “Please let me go, pls, pls, pls.”

When it appeared that he would not be sent to Pakistan, he turned his rage toward France, writing on Feb. 19, 2011, “With Allah’s will, we will find a way to raise the flag here.” A week letter, he wrote that his cell would “wipe away the infidels.” …

Considered the group’s least influential member, Mr. Abballa spent more than two years in prison and was released in 2013. He was kept under surveillance until the end of 2015. …

“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. …

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. …