Not at all, that's a strawman. What you'd conclude is that you have a split predicate, with a subject in between parts. If you claim that "can eat" in "I can eat the cookie" is the predicate, and you also claim that the predicate must be a single consecutive phrase, you'd have a hard time explaining German and Dutch, that "infix" the object: "ik kan een koekje eten".
JAL