We report a search for the anomalous production of events with multiple charged leptons in pp[over] collisions at sqrt[s]=1.96 TeV using a data sample corresponding to an integrated luminosity of 346 pb(-1) collected by the CDF II detector at the Fermilab Tevatron. The search is divided into three-lepton and four-or-more-lepton data samples. We observe six events in the three-lepton sample and zero events in the > or =4-lepton sample. Both numbers of events are consistent with standard model background expectations. Within the framework of an R-parity-violating supergravity model, the results are interpreted as mass limits on the lightest neutralino (chi[over](1)(0)) and chargino (chi[over](1+/-) particles. For one particular choice of model parameters, the limits are M(chi[over](1)(0)>110 GeV/c2 and M(chi[over](1+/-)>203 GeV/c2 at 95% confidence level; the variation of these mass limits with model parameters is presented.