  Distance-based approaches in phylogenetics such as Neighbor-Joining are a
fast and popular approach for building trees. These methods take pairs of
sequences from them construct a value that, in expectation, is additive under a
stochastic model of site substitution. Most models assume a distribution of
rates across sites, often based on a gamma distribution. Provided the (shape)
parameter of this distribution is known, the method can correctly reconstruct
the tree. However, if the shape parameter is not known then we show that
topologically different trees, with different shape parameters and associated
positive branch lengths, can lead to exactly matching distributions on pairwise
site patterns between all pairs of taxa. Thus, one could not distinguish
between the two trees using pairs of sequences without some prior knowledge of
the shape parameter. More surprisingly, this can happen for {\em any} choice of
distinct shape parameters on the two trees, and thus the result is not peculiar
to a particular or contrived selection of the shape parameters. On a positive
note, we point out known conditions where identifiability can be restored
(namely, when the branch lengths are clocklike, or if methods such as maximum
likelihood are used).
