Long time coverage and high radial velocity precision have allowed for the discovery of additional objects in known planetary systems. Many of the extrasolar planets detected have highly eccentric orbits, which raises the question of how likely those systems are to host additional planets. We investigate six systems which contain a very eccentric (e > 0.6) planet: HD 3651, HD 37605, HD 45350, HD 80606, HD 89744, and 16 Cyg B. We present updated radial velocity observations and orbital solutions, search for additional planets, and perform test-particle simulations to find regions of dynamical stability. The dynamical simulations show that short-period planets could exist in the HD 45350 and 16 Cyg B systems, and we use the observational data to set tight detection limits, which rule out additional planets down to a few Neptune masses in the HD 3651, HD 45350, and 16 Cyg B systems.Based on observations obtained with the Hobby-Eberly Telescope, which is a joint project of the University of Texas at Austin, Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen.