  An upper bound on the amplitude of the primordial gravitational wave spectrum
generated during ultra-violet DBI inflation is derived. The bound is
insensitive to the form of the inflaton potential and the warp factor of the
compactified dimensions and can be expressed entirely in terms of observational
parameters once the volume of the five-dimensional sub-manifold of the throat
has been specified. For standard type IIB compactification schemes, the bound
predicts undetectably small tensor perturbations with a tensor-scalar ratio $r
< 10^{-7}$. This is incompatible with a corresponding lower limit of $r > 0.1
(1-n_s)$, which applies to any model that generates a red spectral index $n_s
<1$ and a potentially detectable non-Gaussianity in the curvature perturbation.
Possible ways of evading these bounds in more general DBI-type scenarios are
discussed and a multiple-brane model is investigated as a specific example.
