  We study the conditions for Bloch bands to be spanned by symmetric and
strictly compact Wannier states that have zero overlap with all lattice sites
beyond a certain range. Similar to the characterization of topological
insulators in terms of an algebraic (rather than exponential) localization of
Wannier states, we find that there may be impediments to the compact
localization even of topologically "trivial" obstructed atomic insulators.
These insulators admit exponentially-localized Wannier states centered at
unoccupied orbitals of the crystalline lattice. First, we establish a
sufficient condition for an insulator to have a compact representative. Second,
for $\mathcal{C}_2$ rotational symmetry, we prove that the complement of
fragile topological bands cannot be compact, even if it is an atomic insulator.
Third, for $\mathcal{C}_4$ symmetry, our findings imply that there exist
fragile bands with zero correlation length. Fourth, for a
$\mathcal{C}_3$-symmetric atomic insulator, we explicitly derive that there are
no compact Wannier states overlapping with less than $18$ lattice sites. We
conjecture that this obstruction generalizes to all finite Wannier sizes. Our
results can be regarded as the stepping stone to a generalized theory of
Wannier states beyond dipole or quadrupole polarization.
