Many signaling systems execute adaptation under circumstances that require noise attenuation. Here, we identify an intrinsic trade-off existing between sensitivity and noise attenuation in the three-node networks. We demonstrate that although fine-tuning timescales in three-node adaptive networks can partially mediate this trade-off in this context, it prolongs adaptation time and imposes unrealistic parameter constraints. By contrast, four-node networks can effectively decouple adaptation and noise attenuation to achieve dual function without a trade-off, provided that these functions are executed sequentially. We illustrate ideas in seven biological examples, including Dictyostelium discoideum chemotaxis and the p53 signaling network and find that adaptive networks are often associated with a noise attenuation module. Our approach may be applicable to finding network design principles for other dual and multiple functions.