We present the broad-band noise structure of selected anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) in the 2-60 keV energy band. We have analysed Rossi X-Ray Timing Explorer Proportional Counter Array archival light curves for four AXPs and one SGR. We detect that the persistent emission of these sources shows band-limited noise at low frequencies in the range 0.005-0.05 Hz varying from 2.5 to 70 per cent integrated rms in times of prolonged quiescence and following outbursts. We discovered band-limited red noise in 1E 2259+586 only for similar to 2 yr after its major 2002 outburst. The system shows no broad-band noise otherwise. Although this rise in noise in 1E 2259+586 occurred following an outburst which included a rotational glitch, the other glitching AXPs showed no obvious change in broad-band noise, thus it does not seem that this noise is correlated with glitches. The only source that showed significant variation in broad-band noise was 1E 1048.1-5937, where the noise gradually rose for 1.95 yr at a rate of similar to 3.6 per cent per year. For this source the increases in broad-band noise was not correlated with the large increases in persistent and pulsed flux, or its two short SGR-like bursts. This rise in noise did commence after a long burst, however, given the sparsity of this event, and the possibility that similar bursts went unnoticed the trigger for the rise is noise in 1E 1048.1-5937 is not as clear as for 1E 2259+586. The other three sources indicate a persistent band-limited noise at low levels in comparison.