I think that the reason so many fics get abandoned is a few different reasons, really. I remember writing my first book-length story, this is going back over a dozen years ago, and I couldn't finish it. That and four or five other stories that never really got past Chapter 4... and I kept starting to write new stories until I finally started a book and actually finished it. July 28, 2000. I had been working on novels since the mid 1970's and it took me 25 years to finally complete a book. However, by the time I came to BBM and was encouraged by someone familiar with my other fiction to write a fan fic, I was very wary and decided I was only going to write something short and uninvolved, because I did not want the pressure of having to come through for either the fan or other readers to follow up. So I wrote a 10 chapter novelette. But that took almost no time at all, and I found myself with an idea for another story almost immediately. And because this was about at high-tide of the BBM fandom, there were readers EVERYWHERE, and I quickly got a very vocal following when I started posting Taking Chances. I had expect it to be a 10 chapter story, but BOY WAS I WRONG.
However, not everyone, in fact, very few fan fiction writers have written a novel before, or finished one, and I can say that it is much, much harder to finish a book length story than to start one. It is also, often necessary to go back and revise things, particularly when you aren't working from an outline (I never work from an outline, I wouldn't ever keep to it.) Also, much of the impetus for writing a fan fiction story depends upon the feedback and support of readers, and I have seen countless examples of an author stopping dead after getting unexpected feedback from a reader which either makes them too self-conscious to go on, or defensive, or points up difficulties with the plot or characters that the writer does not know how to resolve. Or someone just being plain mean. Or the writer being devastated by criticism and unprepared for their own lack of motivation to complete the story after that criticism.
Very few fan fiction writers simply stop writing. They, like I when I was first starting to write novels back in high school, found a gap in their motivation or inspiration, and deal with it by changing fandoms, going back to being a voracious reader and stop writing for a period of time (I do this as well, since I am an addict-level novel reader) before returning to start work on something new and fresh and leave the old one hanging.
There are some things to look for, however, to determine whether your author is going to finish or leave you hanging. Primarily, if an author has never finished a story before, chances are that author will not finish the current one. A few other handy tips:
1) if the author has completed writing other books of any genre, or other fan fics, chances are he or she will finish the fic you are reading now. 2) if the author posts updates on a regular basis - even if it is only once per month - and keeps to a regular schedule and seems to have time for the writing, then chances are better than even that the fic will be finished. 3) If the fic is receiving a modest or moderate amount of reader attention rather than an enormous fan club of shrieking fans (they DO put pressure on an author that is not necessarily positive in terms of productivity), it is more likely to be completed, because the author is not depending upon audience attention and feedback as a motivator, and just as important, is not getting distracted with "meta" discussions that take away valuable writing time. 4) If the author seems to be devoted to one story at a time, chances are that story is going to get completed. An author who works on more than one fandom simultaneously, more than one writing project, or more than one fic at a time is dividing her creativity, and the competition for that creativity usually creates two incomplete, inferior products rather than one good solid fic. And 5) if the author vanishes for long periods of time and keeps setting deadlines that are being missed, or has done the same vanishing act in the past on other stories, then the author is really not motivated to complete the story and it is less likely to be finished.
That having been said, all fan fiction is a labor of love and inspiration, and there are a whole lot of factors that go into a story getting finished. But I am sure there are those of you who follow the personal blogs of authors whose work you are waiting for to be updated, and they seem to have all the time in the world for other fandoms, other stories they update more regularly, social life, kids, vacations, leaving you to wonder, "so what are we, chopped liver, aren't we ever going to get the rest of that story?" I have had to face the ongoing desire of readers to see my next story or my next chapter, but I console myself by the fact that I made daily updates to some very long, involved fics for almost 2 solid years before I ever slackened my pace, and I completed every story I started, so if I am making people wait a week or two or three weeks for an update, I don't feel like I am "letting someone down." And when I take a vacation from writing to read 15 or 20 books I didn't read during my 2 year fanfic blitz, I won't feel guilty about it. But I'm an experienced author who has experience finishing books - and most fanfic authors aren't - they are not used to marshalling their energies to get to the finish of a story, they might not know how they will finish it, in which case I think it is only fair to release the outline of the story after a reasonable time period and let the fans "own" the ending if they aren't going to get it from the original writer. And they might have personal and social difficulties that make them want to bail out of the story or the fandom very suddenly - fandom is a volatile social environment.