Even though models for incomplete longitudinal data are in common use, they are surrounded with problems, largely due to the untestable nature of the assumptions one has to make regarding the missingness mechanism. Two extreme views on how to deal with this problem are (1) to avoid incomplete data altogether and (2) to construct ever more complicated joint models for the measurement and missingness processes. In this paper, it is argued that a more versatile approach is to embed the treatment of incomplete data within a sensitivity analysis. Several such sensitivity analysis routes are presented and applied to a case study, the milk protein trial analyzed before by Diggle and Kenward (1994). Apart from the use of local influence methods, some emphasis is put on pattern-mixture modeling. In the latter case, it is shown how multiple-imputation ideas can be used to define a practically feasible modeling strategy.