We discovered that this can occur on a Macintosh computer where there is only one user set up. We are not sure exactly why, but if there is only one user set up on the computer, the activation will not "stick" in some instances.
If you are on a Macintosh, try creating another system user, you don't need to login using the new user account, just having it created should allow the activation to stick.
When you first start the extension, it will need to connect to the internet to double check the activation. the activation will not need to be checked until you close then restart Dreamweaver.