  Brauer and Thrall conjectured that a finite-dimensional algebra over a field
of bounded representation type is actually of finite representation type and a
finite-dimensional algebra (over an infinite field) of infinite representation
type has strongly unbounded representation type. These conjectures, now
theorems, are our motivation for studying (generalized) orders of bounded and
strongly unbounded lattice type. To each lattice over an order we assign a
numerical invariant, $\underline{\h}$-length, measuring Hom modulo projectives.
We show that an order of bounded lattice type is actually of finite lattice
type, and if there are infinitely many non-isomorphic indecomposable lattices
of the same $\underline{\h}$-length, then the order has strongly unbounded
lattice type.
  For a hypersurface $R=k[[x_0,...,x_d]]/(f)$, we show that $R$ is of bounded
(respectively, strongly unbounded) lattice type if and only if the double
branched cover $R^{\sharp}$ of $R$ is of bounded (respectively, strongly
unbounded) lattice type. This is an analog of a result of Kn\"{o}rrer and
Buchweitz-Greuel-Schreyer for rings of finite mCM type. Consequently, it is
proved that $R$ has strongly unbounded lattice type whenever $k$ is infinite.
