Nope, they can't. You'll wish you did, though.
Sort of, on the international version, as if you come back to Besaid later on you're assaulted by Dark Valefor. It's quite easy to beat, fortunately.
The only truly missable things I can think of are all the cute little gifts that priests and passers-by give you around the world... any pieces of equipment they give can be customised later anyway, and little things like recovery items and Gil are easy enough to get your hands on.