It is becoming more and more clear that complex networks present remarkable large fluctuations. These fluctuations may manifest differently according to the given model. In this paper we reconsider hidden-variable models which turn out to be more analytically treatable and for which we have recently shown clear evidence of non-self-averaging, the density of a motif being subject to possible uncontrollable fluctuations in the infinite-size limit. Here we provide full detailed calculations and we show that large fluctuations are only due to the node-hidden variables variability while, in ensembles where these are frozen, fluctuations are negligible in the thermodynamic limit and equal the fluctuations of classical random graphs. A special attention is paid to the choice of the cutoff: We show that in hidden-variable models, only a cutoff growing as with can reproduce the scaling of a power-law degree distribution. In turn, it is this large cutoff that generates non-self-averaging.