We have investigated experimentally the simultaneous transmission of low-power (weak) information-bearing and noise signals in a nonlinear transmission line based on magnetostatic waves in the presence of an additional high-power (strong) signal. Our results - the first of their kind - reveal that when the weak-and strong-signal frequencies are close to one another, a region exists for certain values of their mutual detuning Δf > 2 to 5 MHz in which 'anomalous' suppression of weak signals, including noise, takes place, i.e., the amount of signal suppression in the line is larger than it is when no strong signal is present.