The story can be true.
This sentence is grammatical, but probably doesn't mean what you think. In the examples I can think of, it indicates a possibility that the the truth of the story can change:
Please tell me a story. The story can be true, or it can be made up.
You spend the rest of your days living leisurely on a beach. This story can be true, if you invest with Invesco(tm) today!
If you just heard a story and want to express your uncertainty about its truth, you could say
John told me he won the lottery. This story might be true.
It is equally common in my experience to use could here,
John told me he won the lottery. This story could be true.
A strict grammarian might say that this version actually implies some kind of conditional on the truth, like, "the story could be true, if he would turn in his lottery ticket," but this implication is usually ignored.
The weather is nice but it can change later.
This example is correct, but again could would be more common here.
Edit: According to a note in my Webster's New Collegiate Dictionary, "Can expresses primarily positive power of acting...". That is, can expresses a will to do something in addition to the ability or possibility to do it. This is why the word doesn't work well with subjects like "the story" or "the weather": These abstract subjects don't have a will of their own.