Can an entity track what is being done on a laptop with a wifi connection?

Here is a hypothetical question. If a person uses a laptop that is equipped with a WIFI card, and goes to places like, the public library or a school or workplace, and all of these places have unsecured High speed wireless that ANYONE can just plop down and use, is it possible to catch them downloading illegal stuff (I.E. music, movies, , software, etc...) through things like lime wire, vuse, or regular browsers? And if this information could be discovered would an entity be able to know who it was doing it? I.E. Jeff goes to library downloads a ton of music... should he worry about the police coming to his house? I say yes but im no computer whiz, however most other people say no he shouldn't worry. Thanks in advance.