  We present examples of continuous variable (CV) states having high fidelity
to a given target, say $F > 0.9$ or $F > 0.99$, and still showing striking
differences in their physical properties, including classical and quantum
states within the set, separable and entangled ones, or nearly Gaussian and
strongly non-Gaussian ones. We also show that the phenomenon persists also when
one imposes additional constraints on the energy or the squeezing fraction of
the states, thus generally questioning the use of fidelity to assess properties
of CV systems.
