  We perform electrical field effect measurements at 4 K on insulating granular
aluminium thin films. When the samples size is reduced below 100 micrometers,
reproducible and stable conductance fluctuations are seen as a function of the
gate voltage. Our results suggest that these fluctuations reflect the
incomplete self-averaging of largely distributed microscopic resistances. We
also study the anomalous field effect (conductance dip) already known to exit
in large samples and its slow conductance relaxation in the presence of the
conductance fluctuations. Within our measurements accuracy, the two phenomena
appear to be independent of each other, like two additive contributions to the
conductance. We discuss the possible physical meaning of this independence and
in particular whether or not this observation is in favor of an electron glass
interpretation of slow conductance anomaly relaxations.
