  The problem of existence of adaptive confidence bands for an unknown density
$f$ that belongs to a nested scale of H\"{o}lder classes over $\mathbb{R}$ or
$[0,1]$ is considered. Whereas honest adaptive inference in this problem is
impossible already for a pair of H\"{o}lder balls $\Sigma(r),\Sigma(s),r\ne s$,
of fixed radius, a nonparametric distinguishability condition is introduced
under which adaptive confidence bands can be shown to exist. It is further
shown that this condition is necessary and sufficient for the existence of
honest asymptotic confidence bands, and that it is strictly weaker than similar
analytic conditions recently employed in Gin\'{e} and Nickl [Ann. Statist. 38
(2010) 1122--1170]. The exceptional sets for which honest inference is not
possible have vanishingly small probability under natural priors on H\"{o}lder
balls $\Sigma(s)$. If no upper bound for the radius of the H\"{o}lder balls is
known, a price for adaptation has to be paid, and near-optimal adaptation is
possible for standard procedures. The implications of these findings for a
general theory of adaptive inference are discussed.
