I only say "I'm Chinese American" in formal situations, to people I never met before. And there is a political tint of it, because people know I'm Chinese (most of them can see I'm not Japanese nor Korean). When I add one more word, that's stronger and usually for some purpose.
Generally speaking I agree with Miat. Just to add 君子坦荡荡: when I say I'm Chinese, they know I mean Chinese American. Just like people say "I'm Italian" (in this county), nobody will question what his or her citizenship is.
This is a great country (I hope you know by now) who opens up, and we made the decision to buy into such value and we need to play by it.