After seminal work by Taqqu et al. relating self-similarity to heavy tail distributions, a number of research articles verified that aggregated Internet traffic time series show self-similarity and that Internet attributes, like WEB file sizes and flow lengths, were heavy tailed. However, the validation of the theoretical prediction relating self-similarity and heavy tails remains unsatisfactorily addressed, being investigated either using numerical or network simulations, or from uncontrolled web traffic data. Notably, this prediction has never been conclusively verified on real networks using controlled and stationary scenarii, prescribing specific heavy-tail distributions, and estimating confidence intervals. In the present work, we use the potential and facilities offered by the large-scale, deeply reconfigurable and fully controllable experimental Grid5000 instrument, to investigate the prediction observability on real networks. To this end we organize a large number of controlled traffic circulation sessions on a nation-wide real network involving two hundred independent hosts. We use a FPGA-based measurement system, to collect the corresponding traffic at packet level. We then estimate both the self-similarity exponent of the aggregated time series and the heavy-tail index of flow size distributions, independently. Comparison of these two estimated parameters, enables us to discuss the practical applicability conditions of the theoretical prediction.