I’ve started playing soundtracks to help me relax after work. Most of the time I’ve never seen the movie and have no idea what it’s even about.
But I like the soundtrack. 
this playlist has over 30 recent soundtracks. I’ve seen maybe three of the movies.
Do you only know some movies by their soundtrack?
I won’t answer generally for yes or no, but I enjoyed the original cast recording of Jesus Christ Superstar intensely for years before I bothered watching the movie or seeing it performed live.
Sometimes I’ve gotten surprised watching a movie and suddenly realize that hey! I got that music on my mp3 player. 
I’ve only seen snippets of The Harder They Come, but the soundtrack ( which I have variously owned in cassette, vinyl and CD formats ) is rightly considered a classic introduction to reggae. It made me into an instant fan of (Toots and )The Maytals and Desmond Dekker.
Generally no. There are some exceptions, of course - mostly musicals. But that’s cheating.
Sometimes a great soundtrack can stand on its own. But quite often it’s so tied to the action of the movie that it’s kind of pointless on its own.
Examples: Most of John Williams. I find that his soundtracks usually have a few stand-out pieces, but the rest is just uninteresting – though it may perfectly suit the points of the films it was written for.
I enjoy and regularly listen to several sections of the soundtrack to The Mission, even though I’ve never seen the movie. But that’s a pretty famous score that’s frequently used in advertising and as background music, so I’m probably not the only one to be familiar with the music but not the movie.
I voted “other” because I can think of only one soundtrack* I enjoyed before I ever saw the movie (and I didn’t end up liking the movie much, but still enjoy the soundtrack): O Brother, Where Art Thou?
But that soundtrack, like certain others, is a collection of songs. I can’t imagine buying a “background music” type of soundtrack without seeing the film first, unless maybe it was by an artist I like from the non-movie music world (Stewart Copeland, Jonny Greenwood…)
*Not counting musical documentaries like Gimme Shelter, which has a sort-of soundtrack in the form of the Stones album Get Yer Ya-Ya’s Out!, which I loved long before I saw the film (which I also love).
In a few cases, sure! For me, one example is Chariots of Fire. The music is fantastic, but I don’t really want to see the movie.
I can’t really think of any movies off the top of my head, but it’s happened with TV. I watched two episodes of .Hack//Sign on Adult Swim, hated every second of that boring, plodding, going-nowhere mopey plot with endless panning over scenery as a substitute for actual content (admittedly I may have just seen two bad episodes). Basically, I can’t fairly say I’ve “watched it” in any meaningful capacity.
I also have all the music from all the .Hack series on my iPod because it’s just that damned good. To the point where I considered watching another series recently just because it had the same composer.
For movies, I greatly enjoy Thomas Newman and Rachel Portman. James Horner used to be on that list, but somewhere I crossed a line where everything sounds like something he’s done in the past. Newman’s been flirting with that line for the last couple years, but hasn’t managed to cross it yet.
For TV, I like Jeff Beal’s work.
For games, though he’s only done two, Tomáš Dvořák / Floex (non-game music). In fact, the music he did for Machinarium and Samorost 2 was why I looked up what else he’d done.
For anime: Yoko Kanno, Taniuchi Hideki (the less popular of the Death Note composers, sadly), Susumu Hirasawa (more for the Satoshi Kon stuff than Berserk) and Kunihiko Ryo/Yang Bang-Ean.
Anime’s the longest list since I’m less likely to watch that. There’s more games where I like the music, but I know it from playing the game. Machinarium’s really the only one where I heard the music before I could play the game.
Music is music. I even listen to operas/operettas in languages I don’t know, and have no idea what they’re saying. It’s all good.
I have lots of film music for movies I haven’t seen. Or in some cases, I intentionally hunted down the movie simply because I loved the score so much. The most notable example is John Debney’s amazing score for Cutthroat Island, some of the best film scoring of the 90s. The problem is that (a) the movie is beyond terrible, and (b) all the foley and mix work drown out some of the best cues. Still, great music is great music and needn’t always be so context dependent.