I've heard it said that once is coincidence, twice is happenstance, and three times is enemy action. To say it another way, if in a work of fiction something happens often enough it's preferable for it to have been planned that way by an actor within the story. When writing, I think that keeping one eye on this is important. Good plots should be moved by the choices of the characters who are caught up in them, and not by unlikely intercessors streaking out of the sky. There are very few instances in history where the lack of a coincidental occurence would have been enough to make events switch to an entirely new track.
I remember when I was first starting out with serious attempts at writing, twelve and a half years ago. Back then it was extremely tempting to make things happen just because, without rhyme or reason, so that I could do whatever I wanted and wouldn't have to justify it. Thankfully I left that particular habit by the wayside before I started sending submissions out to the markets in earnest. In retrospect, it's an attitude that works in a similar way as training wheels on a bicycle. With "easy mode" plotting, where things happen because Author Wants Them To Happen, beginner authors have the opportunity to assemble whole stories and start getting handles on the process. Once you have your sense of balance, though, those training wheels will only slow you down and trip you up.
There are probably authors who eschew the use of coincidences as much as possible - I can't think of any offhand, but I would be surprised if they didn't exist. If that's what a person is comfortable with, great. I can understand why some authors would want to banish that kind of capricious artificiality from their stories, so that the plot unfolds solely due to the actions of the characters. I don't go that far. There is room for coincidence in stories - and it helps get a better handle on characters. It opens up their fields of choice, because most people don't factor bolts from the blue into their everyday lives.
Despite the endurance of the old saw that says "there is no such thing as coincidences," I don't agree. I don't think there is any sort of plan for the universe. Things can, and will, happen that individuals just can't predict. That precise thing happened here in Toronto just yesterday. On Tuesday, the Toronto Transit Commission approved a fare increase, one which has resulted in many gnashing teeth and complaining complainers in the media, on Twitter, and elsewhere. On Wednesday, a third-party contractor putting in natural gas lines accidentally broke through into the subway tunnel, which the TTC shut down for a safety inspection - right through rush-hour. People were irritated at having to wait for shuttle buses, already frustrated at the 2010 fare increase.
Coincidences happen, and they can't be predicted. Every once in a while it's useful to drop one on your characters and see how they dance.