  Electrolyte theories enable to describe the structural and dynamical
properties of simple electrolytes in solution, such as sodium chloride in
water. Using these theories for aqueous solutions of charged nanoparticles is a
straightforward route to extract their charge and size from experimental data.
Nevertheless, for such strongly asymmetric electrolytes, the validity of the
underlying approximations have never been properly challenged with exact
simulation results. In the present work, well established mesoscopic numerical
simulations are used to challenge the ability of advanced electrolyte theories
to predict the electrical conductivity of suspensions of charged nanoparticles,
in the salt-free case. The theories under investigation are based on the
Debye-Fuoss-Onsager treatment of electrolyte transport. When the nanoparticles
are small enough (about one nanometer large), the theoretical results agree
remarkably well with the simulation ones, even in the high concentration regime
(packing fraction of in nanoparticles larger than 3$\%$ ). Strikingly, for
highly charged nanoparticles, the theory is able to capture the non-monotonic
variation of the ratio of the electrical conductivity to its value at infinite
dilution (ideal value) as a function of the concentration. However, the tested
theories fail to describe the conductivity of suspensions containing larger
nanoparticles ({\em e.g.} of diameter $4$~nm). Finally, only small charged
nanoparticles can be considered as {\em ions}, as far as electrolyte theories
are concerned.
