Many private ISP's do not change your IP all that often, even if you do have a Dynamic IP Address. You may be able to use your computer daily -- turning it off over night, even -- for weeks without the IP address ever being diffferent. Having a Dynamic IP can often mean that you'll get the IP address you had last time you logged onto the ISP's network unless it isn't available for some reason. If such a reason never arises, you can end up with the same IP address almost indefinitely.
Now, if you have an IP address and several dates/times, you can subpoena the ISP to provide the server logs that relate to that IP address for a window of time encompassing the activity shown in your records. Among those server logs are SMTP logs -- email records of what went out to what addresses and when.
Once all that is known, the noose gets real snug real quick.
For technically minded FReepers, the following is provided:
IP's are tied to MAC addresses which are tied to specific network cards(NIC). Unless someone swaps out their NIC, it is easy to trace the time/date and verify the actual PC. Even if they do swap the NIC, most businesses have with data circuits have a static range of IP's they get assigned. If it was a DSL connection then the ISP can easily tell which phone line was used, and limit the scope of the search to a specific network within a house or business. If the connection was from a cable modem then the range is geographically assigned which means you can tell within a couple of blocks where it came from.