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Source: https://writers.stackexchange.com/a/18822
License name: CC BY-SA 3.0
License URL: https://creativecommons.org/licenses/by-sa/3.0/

Several good answers and I don't want to repeat what they said, but let me add:

I think you can almost always get away with ONE coincidence that gets the story rolling. Like suppose the story begins with two people who are fierce business rivals just happening to meet in a social setting and becoming romantically involved, only discovering their identity as business rivals later. (The Shop Around the Corner, You've Got Mail) Is this an unlikely coincidence? Yes. But few readers will question it, because we accept that this is what gets the story started. It's unlikely, but if it didn't happen, if the two never met, there would be no story at all. When we are relating true stories, we don't talk about the 1000 times that someone went to the grocery store or the office or the park and nothing interesting happened and he went home; we talk about the one time that something interesting DID happen that changed his life. Same in fiction.

Ditto Lauren's comments about turning coincidences into probabilities. If you tell us that one day Bob is walking down the street and he just happens to see a gun lying on the sidewalk and he picks it up and sticks it in his pocket, and then he walks into a bank just as it is being robbed and he heroically saves the day, the reader will likely find the coincidence implausible. But if you tell us that Bob has been carrying a gun constantly every day for the last 20 years, it doesn't sound so implausible. Or if you say that Sally has long dreamed of opening her own restaurant but never had the resources to do it, and one day she attends a class on French literature where she meets someone with a similar dream who is looking for a partner, it sounds like an implausible co-incidence. But change it to a class on how to start a business, and it doesn't sound unlikely at all. Etc.

Original score: 2