The story behind "Never Say Never Again" is long and complicated, but here's the short version:
Producer Kevin McClory had the rights to Fleming's "Thunderball" novel, but found it impossible to mount his own James Bond film in competition with the EON series (this being the mid-60s). He entered a partnership with Broccoli and Saltzman and together they made "Thunderball" in 1965. The understanding was that after ten years, the rights would revert to McClory again (who knew?).
In 1975, McClory appeared from the Phantom Zone and announced he was ready to make his Bond movie. Legal action ensued and finally by 1983 the path was cleared for a new, rival Bond film. Sean Connery had already given his verbal commitment to the film, which by this point had gone through almost as many permutations as the latest Superman movie. One of the unused scripts was even written by Sean Connery in partnership with Leigh Brackett (wife of Superman scribe Ed Hamilton) and involved an A-Bomb under the Statue of Liberty, and mechanical sharks in New York Harbor
. The script that was finally used, for legal reasons, amounts to a pretty direct remake of "Thunderball."
Why did he sign on? Partly as a poke in the eye to his old bosses (when Johnny Carson asked him, on the "Tonight Show," who the villain of the film was, Connery answered, "Cubby Broccoli") and partly, let's be honest, because Sean's career was off the rails by 1983, after a series of flops like "Wrong Is Right," "Cuba," "Meteor," "The Next Man" and so on. Playing Bond again put him back on the map and, once signed to Creative Artists Agency, on the path to the Oscar.
"Never Say Never Again" is not considered an "official" Bond film, as it was made outside the EON-produced series and lacks signature elements like John Barry's "James Bond Theme," the gunbarrel logo, a pre-credits sequence, Maurice Binder-like opening titles, and so on. In fact it was released in competition with an official Bond film, Roger Moore's "Octopussy." However, the rights to NSNA were acquired by EON about five years ago in a legal battle, and so it's at least theoretically possible the film could be included in future collections of Bond DVD's, etc.
The name of the film was suggested by Connery's wife as an "in joke." Ten years earlier, when asked after "Diamonds Are Forever" when he return to the Bond role, Sean had said, "Never again."