Yes, it is possible for someone to copy your copyrighted design and try to claim a copyright on it themselves. If you have not registered your design with the U.S. Copyright Office, you would need proof that the design is rightfully yours with evidence of first dates of display or first sales, or anything that can show you produced the work earlier than the person trying to claim rights in the work. You must have evidence that the person copying the design has infringed on your work.
There are several defenses that can be used if someone is accused of infringement. They can say that they had no idea that the work was copyrighted or they can say that they created the work independently and that it was not copied from the original at all, among others.
One of the ways an artist can protect his/her work prior to registering it (which is not a necessary action) is to place a copyright notice on the work itself - using the copyright "bug", the date of publication, and the artist's name. This gives others notice of its ownership.
One of the advantages to registering any copyrighted work is that it shows the date of registration and, with registration, there is a presumption of ownership in the work.