It is shown that if an economy's participants cannot be separated into groups across which there are no potentially conflicting interests--i.e., if the economy is "indecomposable"--then every continuous truth-dominant allocation mechanism will attain nonoptimal allocations on an open dense set of preference profiles. Classical "Edgeworth-box" exchange economies (economies with no externalities and no production, but with arbitrary numbers of consumers and goods), as well as economies with public goods and economies with other kinds of externalities, are all shown via simple arguments to be indecomposable. The results are extended to cover nonrevelation mechanisms that have dominant-strategy equilibria.