Two felony charges in one day were more than a bump in the NFL’s offseason. They pointed to an ongoing problem for the league — players who wind up at the center of criminal cases.

Patriots tight end Aaron Hernandez was arrested Wednesday in Massachusetts, accused of murdering his friend Odin Lloyd. Also Wednesday, Browns rookie linebacker Ausar Walcott was charged with attempted murder in New Jersey.

Both players were cut later in the day by their teams. On Thursday, the league said any club that now wants to sign Hernandez will face a hearing with Commissioner Roger Goodell first.

The question now is whether the veteran tight end and the rookie should have been in the league at all.

“It is difficult, it’s always a balancing act,” says Tony Dungy, who won a Super Bowl as Colts coach and has served as a mentor to players since leaving the NFL, including Michael Vick after the quarterback served federal prison time for dogfighting. “The league has a security department that sends out information, and every team is different in terms of how much its scouting department does and what areas are concentrated on most.

“It’s really a matter of what you do with the information and what your organization feels is important. One thing you have to keep in mind is a lot of the (negative) things that happen come when they are 15 or 17 or 19 years old.”

According to FBI statistics cited by the league, the incidence of NFL players getting arrested is much lower in than the general public. The average annual arrest rate of NFL players is roughly 2 percent of about 3,000 players who go through the league each year, including tryouts and minicamps. That’s about half the arrest rate of the general U.S. population, the league says. The NFL notes the disparity becomes even more dramatic when the group is narrowed to American men ages 20-34.

But Jeff Benedict, author of several books on athletes and crimes, including “Pros and Cons, The Criminals Who Play In The NFL,” believes the FBI statistics are a bad gauge.

“The danger of doing comparisons with the general public is, if you look at these people and their backgrounds, how many of those guys who have been arrested in the FBI numbers have been to college, make a lot of money like NFL players do, and live in safe, good neighborhoods?” Benedict says. “The issue is why any of these guys are doing this when they have all these good things going on in their lives.”

The San Diego Union-Tribune, which has tracked NFL arrests “more serious than speeding tickets” dating back to 2000, has listed 36 this year, including Hernandez and Walcott and three players who were charged twice.

By comparison, the NBA says six players of its players have been arrested since last July 1, and Major League Baseball says it’s aware of three cases this year worse than a speeding ticket: two DUIs and a misdemeanor drug charge.

While granting that NFL rosters are far bigger than those in the NBA or MLB, Benedict says, “You can’t take these tiny snap shots and say the NFL is low.”

Of course, even a few cases such as Hernandez’s or that of Jovan Belcher — the Kansas City player who shot his girlfriend to death last December, then committed suicide in front of his coach and general manager — can create a widespread negative image.

And anyone who has suited up for an NFL team will face extra public scrutiny for even minor transgressions.

That, in turn, puts more pressure on the league’s vetting process.

Dungy stresses that the amount of homework teams do is critical because they don’t get all that much one-on-one time with prospective players. Some clubs do psychological analyses, even hiring outside agencies to handle them. Though others like the approach, Dungy is not a fan of it and always believed in his gut feeling about a player.

“You have to find out if they have grown from the issues, or there seems to be a pattern, or will these issues always be there,” he says.

Bill Polian, who built the Bills, Panthers and Colts into Super Bowl teams as one of the NFL’s most successful general managers and team presidents, strongly maintains that the league’s vetting process is solid. It delves into players’ histories from high school and college before they enter the league. Those investigations have become more sophisticated through the years; background checks include not only public records such as court documents and arrest data, but talking to teammates and coaches, high school principals and other people who have been a part of a player’s life and development.

“It uncovers a fair amount of information,” Polian says. “It is not designed to uncover information that is usable in court, but it is a process by which the clubs try to ascertain a clear picture of the individual that they are thinking about taking.”

But there’s no way of knowing how playing football for a living will change a young man.

“First of all, it is important to remember that no team is immune from having a player run afoul of the law, whether it’s a speeding ticket up to what we have seen in the Hernandez and Belcher cases, which are as serious as is possible to be,” Polian says.

“There’s no magic wand a team can wave and change that player who has had serious problems. It’s no different than any other workplace in America, just more publicized.”

Benedict agrees that teams perform due diligence on draft prospects and they know what they are getting — or avoiding — in their draft rooms when it comes to skills or 40-meter dash times or health issues.

“The hardest thing they deal with on draft day is the character question,” he says. “That is what keeps them up at night.”

Former Broncos general manager Ted Sundquist says the vetting process wasn’t particularly thorough for many of his years in the NFL — he left after the 2007 season — but he’s certain it is more efficient now.

“I think Michael Vick was the turning point on the timeline,” Sundquist says. “He was right in their backyard and they didn’t know it was going on, an example of a team that had not had a handle on what players were doing.”

Sundquist believes teams could get a better handle on developing problems by hiring security firms that are available around the clock to keep watch on players already in the league, even though the NFL’s personal conduct policy is very direct in saying it expects “lawful, ethical and responsible” behavior.

“It’s better to have a system in place that can monitor or check that guy, a security firm that is part of these guys’ lives, not just vetting them,” he says. “They are tied to the hip with these guys. I think that investment is well worth it.”

Ultimately, if the public grows tired of player misconduct, regardless of the low percentages, it could become a huge problem for the NFL. And it could change how the teams approach player procurement.

“As these issues become a much more public situation in a business that relies upon the public for its goodwill,” Polian says, “you are more and more concerned about taking chances on individuals — no matter what the talent — if they have problems in their background.”

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AP Sports Writers Dennis Waszak Jr., Brian Mahoney and Ronald Blum contributed to this story.