Has any Bond every rolled into the next, or are they all separate stories?
No, all stories have mostly been individual stories; even a producer has stated that Bond is a
series with a series (Whatever that means). But there are elements that have carried over, like Diana Riggs character. Lazenby's Bond marries her in Her Majesty's Secret Service, she is killed and Sean Connery's Bond tracks down Blofeld for revenge in Diamonds are Forever. I think Roger Moore references her in The Spy Who Loved Me and For Your Eyes Only. I think she is slightly referenced by both Dalton's and Brosnan's Bond, but not by name. Some of the props as well are referenced within each Bond and have more or less been a, you know, thing.
I think the same happened in the last Bond didn't it? With Eva Green's character? Because they seem to have gone down the route of trying to cannonize a series, in that, one Bond film, leads to another. This again, is why I posted a massive rant about the overarching storyline, which I now think could be a wrong assumption, but you never know.
There has always been a "Bond" theory that 007 is his sort of "Sort Code" and his name James Bond is in fact, his codename (Or Account Number). This is grasping a bit I think, but does make a bit of sense in the wider picture. Each Bond, is in fact, within the narrative, a different person, but given the Sort Code/Account Number as an undercover name. This would follow in line with M, Q and Moneypenny. Much like how Doctor Who works, each agent retiring and the name living on. There are of course, facts that disprove this theory, but again, I don't buy into that theory much myself. However, the Director of Die another Day is a big proponent of this theory as well as other people attached to the series. (Well, they would wouldn't they?)