  We study an ${\cal N} = 2$ supersymmetric generalization of the
three-dimensional critical $O(N)$ vector model that is described by $N+1$
chiral superfields with superpotential $W = g_1 X \sum_i Z_i^2 + g_2 X^3$. By
combining the tools of the conformal bootstrap with results obtained through
supersymmetric localization, we argue that this model exhibits a symmetry
enhancement at the infrared superconformal fixed point due to $g_2$ flowing to
zero. This example is special in that the existence of an infrared fixed point
with $g_1,g_2\neq 0$, which does not exhibit symmetry enhancement, does not
generally lead to any obvious unitarity violations or other inconsistencies. We
do show, however, that the $F$-theorem excludes the models with $g_1,g_2\neq 0$
for $N>5$. The conformal bootstrap provides a stronger constraint and excludes
such models for $N>2$. We provide evidence that the $g_2=0$ models, which have
the enhanced $O(N)\times U(1)$ symmetry, come close to saturating the bootstrap
bounds. We extend our analysis to fractional dimensions where we can motivate
the nonexistence of the $g_1,g_2\neq 0$ models by studying them perturbatively
in the $4-\epsilon$ expansion.
