If you are going to have a coincidence, I think you should have it at the beginning of the book. I'm reading Elise Broach's Desert Crossing right now. It's a great book. (Even though it's a thriller, there's one kind of funny twist near the beginning that I didn't see coming AND made perfect sense.) In her book, the coincidence is that some kids driving in the desert hit an animal, go back to find it, and find a dead girl instead. it's a desolate place, so what are the chances they would? But because it falls at the beginning of the book, I'm willing to suspend disbelief.
Or look at Nick Hornby's latest. Four strangers meet on New Year's Eve when they all go to the same tall building to commit suicide at midnight. It's a bit of a stretch (and clearly a way for the author to bring together four completely disparate characters), but I bought it. (Full disclosure: I will admit I gave up on the book because I didn't like like three of the four characters, but that was much further down the road.)
I've got a review coming out soon in the Oregonian for a book that's been getting high praise. I won't say the name here, not yet, but it was an excellent book marred by waaaay too many coincidences and cheats. Like we follow the lives and thoughts of many characters, including two who have big, dark secrets that they don't think of at all when we are in their heads. That allows the author to pull off some big surprises (including the story's end), but the twist was a cheat. Other cheats included a murder method (chosen by an assassin) that wouldn't work according to two doctors I consulted. And two characters who are not related to each other but look so much alike they could be identical twins (allowing one person to appear to be alive and dead at the same time). And one more irritation was ending a chapter with a cliffhanger - a character is facing a murderous attacker with no way out - only to reveal several chapters later that the attacker simply dropped dead in the middle of an attack.
I think Ron was right. You can have one unbelieveable thing. One. That's all. And for best results, do it at the beginning.