Disturbance observer (DOB) is one of the most popular robust control tools due to its simplicity and efficiency. The robustness of a DOB based control system changes significantly by the dynamic characteristics of the DOB's low-pass-filter (LPF) and nominal plant. The bandwidth of the LPF of a DOB is desired to set as high as possible to estimate/suppress disturbances in a wide frequency range; however, the practical and robustness constraints limit the bandwidth of the LPF of a DOB. This paper clarifies the robustness constraints of the DOB's bandwidth by using two different robustness analysis methods, i.e., real parametric uncertainty and unstructured uncertainty based analysis methods. It is shown that the bandwidth of a DOB has a lower bound to obtain robust stability, and the stability margin improves as the bandwidth of the DOB is increased when the plant includes real parametric uncertainties and the order of the DOB is one. The robustness analysis is extended into non-minimum phase plants by using unstructured uncertainty based analysis methods. It is shown that the bandwidth of a DOB has upper and lower bounds to obtain a good robustness if the plant has non-minimum phase zero(s) and pole(s), respectively. Besides that the order of a DOB has a design trade-off between the robustness and performance: if a higher order DOB (HODOB) is used, then the bandwidth constraints of a DOB become more severe, yet the performance is improved. As a result, new analysis and design methods are proposed for the DOB based robust control systems. The validity of the proposals are verified by simulation results.