Yes, you can. Usually, the serial number is on the back of the cd-case or on the back of the installation guide. If you didn't get one, then either you have an illegal copy of the game, or you should go back to where you bought it and ask for it. If you have an illegal copy, then you should find your nearest computer-store and buy the game. There's no other way to get a serial number in that case.
There are 3 ways to get your original serial number back. First of all, you can use the Windows registry to get it back. The easiest way to use the registry is by installing and using The Add-on Handler from the Utilities section on this site. You'll need at least version 0.3.0beta. After installing and running that, go to the "Serial". There you will see all the serial number for the Sims games you installed. Note that The Sims Online is not included with that.
If it's not in the registry, but you registered the game on the official site then you can get the serial number back from there. Go to "Register" at the top of the official site
and go to Product Registration. Click "Add serial numbers" to find your serial numbers there.
Always register the game on the official site, so you can always retrieve your serial numbers from there!
As last option you can call EA Technical Support. The number or email address for your country can be found in the installation guide, or on the EA Website of your country. EA will be able to replace your serial number, usually for a small fee. See www.ea.com
to find the EA website of your region. If you're in the US, see this page
for more help. UK/Ireland residents can check this page
if the serial on the manual is wrong/misprinted, or this page
if the serial number has been lost.
If EA has no phone number in your country, find the technical support section on the EA website, and try to ask a question there. You should be able to get in touch with EA support personnel through there.