With dial up, you’ll get a different IP addr every time you connect.
You’re liable to also even with a cable modem, provided you actually shut down the modem between sessions.
You’re liable to also even with a cable modem, provided you actually shut down the modem between sessions.
However, your Internet Service Provider needs to know who you are when you log in - to verify that you are a valid bill-paying customer. To accomplish this, they give you a sign-in and a password (or at the very least, they look at the unique MAC address on your cable modem to verify that it is you). Either way, once you connect to your service provider, THEY know who you are, even if the government does not.
Once you successfully connect to your service provider, and sign in, THEN they assign you an IP address, which you then use for surfing the Web. Even if you hang up, log back in the next day, and get a different IP address (assuming you have dial-up), your Internet Service Provider will still have a record of which IP address they assigned to you the previous day (and probably every day before that).
Hence, all anyone with authority has to do is go to your Internet Service Provider and ask a question like: "on 10/14/08, at 11:23pm, which one of your customers was assigned IP address 43.67.21.56??". So I don't think that hanging up and redialing, or power-cycling your cable modem, will do much good if they are sufficiently determined to identify you.