As a general rule of thumb, having *something* in your plot happen as a coincidence is a bad idea. It's just not very dramatic and *can* lead to the dreaded Deus Ex Machina, especially if your protagonist needs some help on something, whether that's escaping jail, a maniac killer or the attentions of an overly-enamoured school teacher (or whatever, stay with me).
However, we all know films and TV drama use coincidence all the time and get away with it. So what's the difference?
Well, it seems to me that if your coincidence gets your characters OUT of trouble, then you will be found out... But if your coincidence gets them INTO trouble, you *may* just get away with it.
But please, please, please don't have your characters meet by coincidence... This seems to be the biggest use of coincidence in the specs I read and I am struggling to think of one that has worked:
CHARACTER 1: Wow, fancy seeing you here.
CHARACTER 2: I know! I haven't seen you in like, forever.
CHARACTER 1: Yeah, what are the odds? Now we have the excuse to get together and go on some kind of adventure, romp or octane-fuelled horror ride where we're hunted through the woods like animals.
EESH. It *feels* constructed. As do the characters who have never met before, but *bump* into each other:
CHARACTER 1: Hey! Look where you're going, will you?
CHARACTER 2: I'm so sorry... Can I buy you a drink? Y'know, just to say sorry.
CHARACTER 1: Okay, sure person I've never met before in my life who could be an axe murderer. Because unlike in real life where I'd just keep walking, I'll now stay here and get involved in your crazy schenanigans or let you kidnap me and sell me on the black market like the Luc Besson movie, TAKEN*.
So whilst coincidence *can* be okay, most of the time it isn't. Sometimes it feels like coincidence in your only option when it comes to plot construction; I've been there, too. But in reality: IT ISN'T. There are always ways of getting your characters where you want them. You're the writer, the puppet master. So get those puppets in the right place, STAT.
*Taken really was a looooooooong string of "unlikely coincidences" as one critic put it... Check out its much superior predecessor, SPARTAN (written and directed by David Mamet) to see how it's really done.
UPDATE: Hmmmm... WHAT A COINCIDENCE! A great post from Cracked.com about INSANE coincidences - you really won't believe some of these. Thanks to @Ditty1013 on Twitter.