on a 56k modem, just restart your computer, to change the IP address.
on broadband, some can, some can't. depends on the ISP.
dial-up services have a limited number of IP addresses (cheaper since not everyone is on at the same time), called a Pool. when you log on to your ISP, you get a random number from the pool.
unfortunately, when dial-up user is banned, usually the IP is broad to cover this.
for Broadband, the ones you have to dial up may have an IP pool just like dial-up.
for always on connections, it is usually a static IP that never changes. thus, the board can just use a specific IP ban.