You can get a new IP address, but it rarely changes... I think it only changes when people join and/or people turn off their modem (Cable or DSL). Turning off the computer wont change the IP address. If you unplug or disconnect your modem from the phone or cable line, then the IP address is released for someone else to take.
ipconfig lists information about the network adapters. ipconfig /renew obtains the IP address from the modem since it is the first network device that connects to the ISP. ipconfig /release is used to "stop" network activity for the computer, but the IP address remains active on the modem.
To get a new IP, look at the DHCP lease expiration. When the lease expires, it automatically renews instantly, probably giving you the same IP again. To prevent that, you have to unplug your modem or disconnect the cable or phone line to the modem for a day or two after the expiration date.
<Edited for Clarification>