  In the recent years self-testing has grown into a rich and active area of
study with applications ranging from practical verification of quantum devices
to deep complexity theoretic results. Self-testing allows a classical verifier
to deduce which quantum measurements and on what state are used, for example,
by provers Alice and Bob in a nonlocal game. Hence, self-testing as well as its
noise-tolerant cousin -- robust self-testing -- are desirable features for a
nonlocal game to have.
  Contrary to what one might expect, we have a rather incomplete understanding
of if and how self-testing could fail to hold. In particular, could it be that
every 2-party nonlocal game or Bell inequality with a quantum advantage
certifies the presence of a specific quantum state? Also, is it the case that
every self-testing result can be turned robust with enough ingeniuty and
effort? We answer these questions in the negative by providing simple and fully
explicit counterexamples. To this end, given two nonlocal games $\mathcal{G}_1$
and $\mathcal{G}_2$, we introduce the $(\mathcal{G}_1 \lor
\mathcal{G}_2)$-game, in which the players get pairs of questions and choose
which game they want to play. The players win if they choose the same game and
win it with the answers they have given. Our counterexamples are based on this
game and we believe this class of games to be of independent interest.
