A few weeks ago, I met someone who had been one of my closest friends as a teenager, but whom I'd not seen for decades. We'd lived in the same town for most of those years, and could easily have bumped into each other shopping, but we didn't. That's probably more remarkable than the fact that when we did meet, in ludicrous circumstances, we still recognised each other.
It was a real coincidence, and if I write a memoir I might put it in, but if I ever tried to base a short story around it, it would probably flop. The reason? Because coincidences always come across as unbelievable and plot cop-outs in fiction. Even fictionalised accounts of history struggle to make coincidences acceptable e.g. when Richard III was crowned there was an eclipse, which in those superstitious times was taken as a bad omen. That was fact, but any reader not knowing this would think it a totally over-the-top creation of the author–too far-fetched to be true.
There are instances when plots have relied on coincidence and got away with it. In Jane Eyre, Charlotte Brontë has the eponymous heroine in dire straits in the middle of nowhere, rescued by people who turn out to be cousins she never knew she had. What are the odds? But Jane Eyre was a ground-breaking novel whose other aspects were far more likely to cause controversy, such as a hero who keeps his mad wife in the attic and attempts bigamy, and a feisty heroine who speaks her mind. Charlotte Brontë was no second-rate author. She created memorable characters and evocative settings, so she can be forgiven for any unwieldiness in her plot. Other authors are unlikely to be as lucky.
I'm not saying that coincidences should be avoided in fiction altogether, only that if you don't want your readers to feel cheated, you need to treat them with caution.