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Extrasolar multiple-planet systems provide valuable opportunities for testing theories of planet formation and evolution. The architectures of the known multiple-planet systems demonstrate a fascinating level of diversity, which motivates the search for additional examples of such systems in order to better constrain their formation and dynamical histories. Here we describe a comprehensive investigation of 22 planetary systems in an effort to answer three questions: (1) are there additional planets? (2) where could additional planets reside in stable orbits? and (3) what limits can these observations place on such objects? We find no evidence for additional bodies in any of these systems; indeed, these new data do not support three previously announced planets (HD 20367 b: Udry et al.; HD 74156 d: Bean et al.; and 47 UMa c: Fischer et al.). The dynamical simulations show that nearly all of the 22 systems have large regions in which additional planets could exist in stable orbits. The detection-limit computations indicate that this study is sensitive to close-in Neptune-mass planets for most of the systems targeted. We conclude with a discussion on the implications of these nondetections.