If you've never read the Lord of the Rings, or you read it a long time ago, then you might not realise that there are 6 books in the series. They're sold in just 3 volumes, called Fellowship of the Ring, The Two Towers, and The Return of the King, but each volume is divided into two books each. The divisions between the books aren't arbitrary, with each one telling a distinct aspect of the LOTR saga. I think the way Tolkien planned his novels is significant, and it's no accident that he wrote 6 books and not just the 3 that were published.
Here are the books Tolkien wrote:
You might notice from these descriptions that Tolkien told the story of the Ring, especially in the latter books, in modules. The story of Frodo and Sam's quest doesn't intersect, at least in the narrative, with the story of Aragorn and Gimli and Legolas. Even though they happen concurrently, they're treated as separate stories. To tie them together, Tolkien provides dates and some key events seen from afar.
I don't know why Tolkien chose to keep the intertwined stories of LOTR separate, but I suspect he was compelled to do it by a storytelling instinct. Whether Tolkien understood it or not (and because he wasn't a full-time author, I suspect he didn't actually realise it about himself), he knew that the world he was building and the fantastic history he was writing was complex. He obviously understands and knows his fictional world and its history, but his readers didn't [yet]. And he knew that.
By keeping the threads of his tale separate from one another, he helped us readers keep track of every major character's whereabouts, motivations, concerns, and threats. Modern storytelling has very much strayed away from this. Today, books are written to mimic visual media. Authors cut between plot lines, and they reap the same benefits from it as movie editors do. They're built-in cliffhangers. The more you "cut" away from one plot, the more anxiety you build in your audience.
It works so well that it almost seems like Tolkien missed an opportunity by not doing it in his books. The Peter Jackson movies reinforce this. The movies cut between the story lines, and it works well emotionally. Whether it works well narratively is another question.
When I read a very complex story like a 300+ page Warhammer novel, or the series of 50+ 300 page novels in the Horus Heresy, I have to admit I often find it difficult to follow all the character names and locations and plotlines. Allowing for the fact that I'm no neurologist and have no idea how brains work, it seems to me that every time a book cuts back to a plot line it's wandered away from, you have to try to reload a set of memories about what's essentially a different story. Then the book cuts away from that plot again, so you discard information and load what memory you've managed to retain about it. Repeat this cycle enough, and by the end of the book you're lucky to understand how anybody got to where they end up.
By recognising that he was actually telling two stories, and organising them into separate books, Tolkien helps you keep track of the progress, and all the minor details, of each major plot line.
I think the language of cinema necessitated a certain pattern of storytelling, so the movies interweave the stories successfully. We're all used to that in movies, and there's a lot of additional visual data to help us keep track of who's who and where they are and what they're trying to achieve. Think of just how often we see the "eye of Sauron" in the movie, or the tower in Isengard, to remind us of which bad guy is which. But a book is a different medium, and Tolkien structured his stories to help his readers comprehend the tale he was trying to tell.