Can an animal be both an arthropod and a vertebrate?

No. Arthropods by definition are invertibrate and have an exoskeletons, or rely upon an external skeleton. Vertebrates like mammals, reptiles, birds, etc., possess vertebrae as part of an endoskeleton (internal). Note that the Word "vertibrate" refers to a type of organism having a spine or backbone (for example, 'a goat is a vertebrate'), whereas "vertibrae" refer just to the bones themselves (for example, 'a goat has vertebrae in its spinal column').