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The biggest threat to your anonymity is the fact that the APs are probably set up to log the MAC addresses of computers that connect. Now, while they may not be able to precisely figure out who you are from that, your MAC address will tell "the man" some information:
What brand of network card you use. If they do decide to run a sting operation against you, they could always set up a triangulation network and wait for you to show up and your MAC address to appear, and then verify that it's you using it.
Of course, that would probably have the pre-requisite of you doing something highly illegal. If your IP address did get busted for P2P sharing, the worst that would probably happen is life would get bad for the coffee shop owner and he'd shut down the AP, or at least make people pay to use it.
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