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We present new spectroscopic observations for a sample of C(N)-type red giants. These objects belong to the class of Asymptotic Giant Branch stars, experiencing thermal instabilities in the He-burning shell (thermal pulses). Mixing episodes called third dredge-up enrich the photosphere with newly synthesized C12 in the He-rich zone, and this is the source of the high observed ratio between carbon and oxygen (C/O > 1 by number). Our spectroscopic abundance estimates confirm that, in agreement with the general understanding of the late evolutionary stages of low and intermediate mass stars, carbon enrichment is accompanied by the appearance of s-process elements in the photosphere. We discuss the details of the observations and of the derived abundances, focusing in particular on rubidium, a neutron-density sensitive element, and on the s-elements Sr, Y and Zr belonging to the first s-peak. The critical reaction branching at Kr85, which determines the relative enrichment of the studied species, is discussed. Subsequently, we compare our data with recent models for s-processing in Thermally Pulsing Asymptotic Giant Branch stars, at metallicities relevant for our sample. A remarkable agreement between model predictions and observations is found. Thanks to the different neutron density prevailing in low and intermediate mass stars, comparison with the models allows us to conclude that most C(N) stars are of low mass (M < 3Mo). We also analyze the C12/C13 ratios measured, showing that most of them cannot be explained by canonical stellar models. We discuss how this fact would require the operation of an ad hoc additional mixing, currently called Cool Bottom Process, operating only in low mass stars during the first ascent of the red giant branch and, perhaps, also during the asymptotic giant branch.