No. As others have pointed out, Applecare is tied to that particular machine (i.e. serial number) not the owner, which is why when you sell the computer (should you choose to do so) Applecare follows the computer.
You can buy Applecare at any time during the first year of ownership, up to the end of the twelfth month, but at no time after that.
Personally, I have always bought Applecare for all of my Mac computers.