Coincidences happen all the time in real life. An ex-colleague recently attended two meetings at an employment consultancy. Both times she chose a seat at random, and both times she found herself seated next to someone recently made redundant within the same industry as her. In a story that kind of coincidence would make a perfect starting point. Three people realise that unexpectedly they have something in common, and that leads to a greater level of connection which in turn drives the plot forward. At the start of a story, it's often a random chance event that throws the protagonist out of their regular life and into the events of the unfolding plot.
Apparent coincidences later in the story may, however, be more difficult for a reader to accept. Sometimes these happen because the author knows more than either the characters or the reader. It can be difficult to convey this information (or at least the existence of the information) when writing in a limited point of view and where there is no way for the characters to know what is going on.
Personally I like the idea that there's a whole big world out there that the central characters can never fully know, and can only influence in small ways within their own areas of expertise. If someone else is pulling the strings, however, it's important to at least hint that to the reader.
I've got that issue in a story I'm revising at the moment. The protagonist has been given a job to do, not knowing that her employer has also hired someone else to carry out a related task in the same time or place. I want the appearance of that second character to be almost as big a surprise to the reader as it is to the protagonist, but it currently looks like a little too much of a coincidence.
My thought is that my protagonist's employer should explain her lateness to the meeting (not alluded to in the current draft) as being because she had a prior engagement to hire someone else (without mentioning what for), who is 'a bit of a loose cannon'. This then hints to the reader (who expects story conventions to be followed, at least to a certain event), who this unseen third party is, and that they may have a further part to play in the story. Meanwhile our protagonist will continue with her mission (not knowing that she's in a story and that plot conventions exist), believing that she is the only employee sent to that location, and that her employer was referring to someone completely unknown to the protagonist.
As a reader, would that be enough of a hint, not to make the later plot events seem like a coincidence of writer convenience?
What coincidences can you, as a reader, accept in a story, and which will make a story completely unbelievable?
What problems have you, as a writer, encountered in trying to bring plot elements or characters together without relying on chance happenings, and how did you overcome the problem?