Nonlinear feedback controllers are ubiquitous features of biological systems at different scales. A key motif arising in these systems is sequestration-based feedback. As a physiological example of this type of feedback architecture, platelets (specialized cells involved in blood clotting) differentiate from stem cells, and this process is activated by a protein called Thrombopoietin (TPO). Platelets actively sequester and degrade TPO, creating negative feedback whereby any depletion of platelets increases the levels of freely available TPO that upregulates platelet production. We show similar examples of sequestration-based feedback in intracellular biomolecular circuits involved in heat-shock response and microRNA regulation. Our systematic analysis of this feedback motif reveals that platelets-induced degradation of TPO is critical in enhancing system robustness to external disturbances. In contrast, reversible sequestration of TPO without degradation results in poor robustness to disturbances. We develop exact analytical results quantifying the limits to which the sensitivity to disturbances can be attenuated by sequestration-based feedback. In summary, our systematic analysis highlights design principles for enhancing the robustness of sequestration-based feedback mechanisms to external disturbances with applications to both physiological and cellular systems.