Blindly, I say it can be somewhat traced by it's MAC number since big manufacturers purchase/reserve MAC numbers in blocks. This allows determining what the laptop is, but not necessarily which individual one it is from a specific company unless they kept really good records of each and every computer made, like Dell and their service tags that seem to know all about your computer.
It gets a bit fuzzy on some individual cards sold separately. Boatloads of e-waste goes to China for "re-processing" and that includes network cards with their MAC numbers within that get re-cycled into a so-called "new" product.
So the final answer is... Yes, we can find you, if you have Java enabled.
Let's suppose someone wanted to misuse the Internet by, say, libelling someone and wanted to hide by posting from an anonymous WiFi, they could be traced after the even when investigators sought the information on who used that IP address at that time and the MAC address would be part of the informaiton given over.
In extreme circumstances, that MAC address could then be traced from future use.