The conductance noise of Anderson-insulating amorphous and crystalline indium-oxide films exhibits a number of intriguing features. In the linear-response regime (small applied fields), the power spectrum (PS) in these systems is 1/f-like and does not exhibit saturation down to f approx 103 Hz. A sharp transition in the character of the noise occurs when the conductance is measured some distance into the non-Ohmic regime, and when the sample is immersed in liquid He. The PS acquires a strong component that is flat up to frequency f*, above which it declines sharply with f (faster than a power-law), and eventually is masked by the 1/f noise. Measurements in the time domain reveal that this anomalous noise is due to downward-going spikes in the conductance G, which start to appear once the drive current exceeds a threshold value. The average frequency of the occurrence of spikes increases monotonically with the current density. The spikes shape varies somewhat from sample to sample but it is usually asymmetric with a faster attack-edge and a slower trailing-edge. The characteristic duration of each spike τ in a given sample increases with the drive-current (causing f* in the PS to increase). More importantly, t depends on the carrier concentration n. For high-density systems (n greater-than-or-equal-to 1021 cm-3) τ may be as high as 200mS near threshold. As n goes down τ decreases and for n less-than-or-equal-to 1019 cm-3 τ is of order of 1mS. The detailed way τ depends on n correlates with the relaxation times in these systems, which are known to be electron-glasses. The anomalous noise is heuristically interpreted as an out-of-equilibrium phenomenon associated with two ingredients: A congested current-flow situation developing at some bottleneck resistor in the percolation network, which in turn results in a short-lived local field across a bottleneck resistor. This field may nucleate a cavity at the sample-helium interface that produces as thermal shock, which is converted by the sample to exhibit a current spike.