  We show that the system of $k$ NS5-branes wrapping $\mathbb{T}^4\times S^1$
has non-trivial vacuum structure. Different vacua have different spectra of 1/4
BPS states that carry momentum and winding around the $S^1$. In one vacuum,
such states are described by black holes; in another, they can be thought of as
perturbative BPS states in Double Scaled Little String Theory. In general, both
kinds of states are present. We compute the degeneracy of perturbative BPS
states exactly, and show that it differs from that of the corresponding black
holes. We comment on the implication of our results to the black hole
microstate program, UV/IR mixing in Little String Theory, string
thermodynamics, the string/black hole transition, and other issues.
