Characteristics of multi-particle production in proton-proton collisions at root s = 7 TeV are studied as a function of the charged-particle multiplicity, N (ch). The produced particles are separated into two classes: those belonging to jets and those belonging to the underlying event. Charged particles are measured with pseudorapidity |eta|< 2.4 and transverse momentum p (T)> 0.25 GeV/c. Jets are reconstructed from charged-particles only and required to have p (T)> 5 GeV/c. The distributions of jet p (T), average p (T) of charged particles belonging to the underlying event or to jets, jet rates, and jet shapes are presented as functions of N (ch) and compared to the predictions of the pythia and herwig event generators. Predictions without multi-parton interactions fail completely to describe the N (ch)-dependence observed in the data. For increasing N (ch), pythia systematically predicts higher jet rates and harder p (T) spectra than seen in the data, whereas herwig shows the opposite trends. At the highest multiplicity, the data-model agreement is worse for most observables, indicating the need for further tuning and/or new model ingredients.