The torproject guys themselves recommend us to put a particular HTTP proxy (Polipo) between our browser and Tor to deal with latency. I guess such proxy does so by opening multiple parallel connections to fetch the different resources it sees on an HTML before the browser even requests for them. That said, maybe multiple connections to the same IP reuse the same Tor circuit....
But anyway, you should protect your bitcoin traffic with Tor or something alike for the reason others have said in this thread.
Look at blockchain.info for example. It already tries to link transactions to IPs. If blockchain.info is one of your peers, every transaction you make will be publicly linked to your IP forever on their site. And it may even accidentally link you to transactions that you relay to them, if they are not connected to the original sender but you are. People may even conclude false things about you, if their accuracy rate ever gets high enough but you happen to be among those "error cases"...