This paper studies limitations of disturbance rejection in closed-loop control systems. The disturbance is modeled as colored noise, and disturbance rejection is measured by the output variance. Lower bounds on the achievable output variance are derived under several practical constraints, the main one being a bound on the system bandwidth; constraints on the stability robustness (measured by the distance from the Nyquist plot to the critical point) and the ability to track low-frequency reference signals (measured by the sensitivity reduction at low frequencies) are also considered. The bounds are derived by finding the sensitivity function that minimizes the output variance. This optimization is performed subject to the above constraints and, in addition, subject to certain sensitivity integral constraints that arise because of closed-loop stability. The utility of the results is illustrated by analyzing the achievable level of passenger comfort in an aircraft subjected to turbulence.