  We analyze the connection between Bell inequality violations and symmetric
extendibility of quantum states. We prove that 2-qubit reduced states of
multiqubit symmetric pure states do not violate the Bell
Clauser-Horne-Shimony-Holt (CHSH) inequality. We then prove the more general
converse that any 2-qubit state that violates the CHSH inequality cannot have a
symmetric extension. We extend our analysis to qudits and provide a test for
symmetric extendibility of 2-qudit states. We show that if a 2-qudit Bell
inequality is monogamous, then any 2-qudit state that violates this inequality
does not have a symmetric extension. For the specific case of 2-qutrit states,
we use numerical evidence to conjecture that the
Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality is monogamous. Hence,
the violation of the CGLMP inequality by any 2-qutrit state could be a
sufficient condition for the nonexistence of its symmetric extension.
