It’s been four years since lawsuits connected to concussions were first filed against the NFL, and even now that a settlement agreement finally has been approved by a federal judge, none of the hundreds of millions of dollars available to thousands of former players will be paid until any appeals are heard — and, after that, until claims are assessed.

“It’s a really good feeling, but it’s not over yet, because of the appellate process,” said Craig Mitnick, who represented more than 1,000 ex-players against the NFL. “It is more likely than not that within the next month, one of the attorneys that got involved late in the game, who represents a player or players who do not believe the settlement is fair, is going to court.”

Co-lead plaintiffs’ lawyer Christopher Seeger said he has “a modest concern” about appeals.

“The most common question we’ve heard from retired players has been: How quickly can we get help?” Seeger said. “As long as no appeals are filed, the settlement process will begin to roll out its benefits as soon as this summer.”

The lawsuits accused the NFL of failing to properly protect players from the long-term risks of concussions.

Senior U.S. District Judge Anita Brody in Philadelphia approved the settlement with a 132-page opinion Wednesday after twice asking lawyers to do more work because of her concerns over whether the initial ceiling of $765 million for payments to players was too low. That figure, first announced in August 2013, was not increased, but the cap was removed. So it’s not known exactly how much the NFL will pay, aside from about $100 million in lawyers’ fees, along with millions more for concussion education and monitoring of ex-players’ health.

“It is an uncapped settlement, so it could be much more” than $765 million, Seeger said.

In theory, it also could be less.

Dozens of ex-players or their relatives rejected the settlement, including the family of Junior Seau, the Pro Football Hall of Famer who killed himself in 2012.

Last year, seven former players who objected to the proposed settlement unsuccessfully brought their complaint to the 3rd U.S. Circuit Court. One, Sean Morey, said no decision had been made about appealing Wednesday’s order.