'Possible', by itself, applies to everything, so you don't say anything by declaring something 'possible'. It's only when you start talking about degrees of possibility that the word has any meaning.

Part of the problem is that "possibility" has different meanings depending on context. In everyday speech, it seems to be used to indicate degrees of probability. When people declare a certain event "possible" in everyday speech, they usually mean that it has a low but nontrivial probability, given the everyday state of the world. In this sense, I might say that it's "impossible" for me to become a NFL player, even though in a philosophical discussion we would recognize that the probability that I could become a NFL player is greater than 0.

Problems occur people equivocate between different meanings of "possibility," or introduce a certain meaning into a type of discussion where it doesn't belong. For instance, it's "possible" that the Flying Spaghetti Monster exists and created the world, but this is not the kind of "possibility" that people deal with in everyday life.