Is it possible for the lungs to inflate/chest to expand, but for no or little air to get into the body?

By laws of nature...: No. If a person's lungs are inflated with air, or if his chest is inflated with air, then by the laws of physics and the laws of nature, he has air in his body. That's the definition of "air in the body". Of course, there may be research into the futuristic breathing of "liquid air" as some propose for deep sea diving or space travel, but that's another story.