Thinking about this topic, I've come to the conclusion that I'm fairly good/lucky at choosing books to read I think I'll be interested in. Since I only read a couple of books a month, that's probably an important skill as well - as others have commented, it's depressing to spend time doing something you don't like for the sake of completeness. I'd add to it though, that some books while not necessarily a good "read" were very thought-provoking and I was glad to have read them.
As for my record of finishing books, the only ones I've never finished were school/course assignments that I usually couldn't finish within the time constraints - usually because I'd keep falling asleep from boredom, or had to read slowly because of comprehension/translation issues. My lack of interest in some of those makes me unsure exactly which ones I haven't read! But I do know that my record is far from perfect.
As far as Tolkien's
TLOTR, I finished that as a teenager (and again later as an adult) and I think the turn-offs that many people cite were actually half of what I liked about the book. I think of myself as very detail-oriented (sometimes losing track of the forest for the trees) and given the completely novel and fantastical setting (or at least what seemed so to me at the time) each new character mentioned or tangential meandering from the main storyline made me feel like I was being let in on some secret knowledge.
Regarding Ayn Rand, I read (and finished)
The Fountainhead my senior year in high school and by the time I reached the rape-but-not-really-a-rape scene, I had a fairly clear understanding of the basic ideas Rand was trying to convey. Finishing the novel was not a problem, as it was the way of finding out what exactly she was trying to say in her allegorical work. Of course, not really agreeing with most of her views, I've never had any inclination to start
Atlas Shrugged.
There was one work that I was extremely surprised was not on the list: The Bible. I know a LOT of people (mostly Christians, of course) who start out intending to read it cover to cover but start losing steam around the "Begattitudes", start wavering around Leviticus/Numbers and then usually give up before Psalms. And that's not even including people who stop reading because they find something that contradicts what they were taught or believed.
As for books that I have finished, the ones I've had the most trouble with finishing were Hugo's
Les Miserables (his digressions make Tolkien's look like hiccups) and a collection of works,
The Slave being the longest, by Isaac Bashevis Singer. I don't think it's coincidence that translated works such as those tend to slow down my reading a lot.