No, you can not. You must purchase it and activate it prior to the expiration of the standard one-year warranty. If you have just missed it by a day or two, you can try to contact Apple and beg them for leniency, but don't hold your breath.
This policy prevents people from buying AppleCare but not activating it with a machine until/unless they have a problem with it. If the machine had no problems before three years were up, they'd still have an unused AppleCare package to sell or apply to another machine. Not good for Apple's bottom line.