Yes they can. Authorities can (and have been known to) upload a fake file that people want. Then, anyone that connects to download it will share their unique IP address with the uploader. The authorities can then track down your ISP using your IP address, request a warrant for the address of the home that gets that IP, knock on your door, and then arrest you.
However, this seldomly happens because, technically speaking, the illegal action is *distribution*, not downloading. But it's a very gray line and people have been prosecuted using this method.
EDIT: Almost forgot one detail. As you're downloading the file, whatever has been downloaded will automatically be shared with others that need that portion. So if you have the first 10 minutes of a movie, while the download is finishing you are sharing the first 10 minutes with anyone who still needs it.
But again, these are just the possibilities. In practice, they don't really care.