Characteristics of multi-particle production in proton-proton collisions at √s = 7 TeV are studied as a function of the charged-particle multiplicity, Nch. 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 |η| < 2.4 and transverse momentum pT > 0.25 GeV/c. Jets are reconstructed from charged-particles only and required to have pT > 5 GeV/c. The distributions of jet pT, average pT of charged particles belonging to the underlying event or to jets, jet rates, and jet shapes are presented as functions of Nch and compared to the predictions of the PYTHIA and HERWIG event generators. Predictions without multi-parton interactions fail completely to describe the Nch-dependence observed in the data. For increasing Nch, PYTHIA systematically predicts higher jet rates and harder pT 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.