Fiducial-based rigid registration is the preferred method for aligning the preoperative image with the intra-operative physical anatomy in existing image-guided surgery systems. After registration, the targets locations usually cannot be measured directly, so the Target Registration Error (TRE) is often estimated with the Fiducial Registration Error (FRE), or with Fitzpatrick TRE (FTRE) estimation formula. However, large discrepancies between the FRE and the TRE have been exemplified in hypothetical setups and have been observed in the clinic. In this paper, we formally prove that in the worst case the FRE and the TRE, and the FTRE and the TRE are independent, regardless of the target location, it location, the number of fiducials, and their configuration. The worst case occurs when the unknown Fiducial Localization Error (FLE) is modeled as an affine anisotropic inhomogeneous bias. Our results generalize previous examples, contribute to the mathematical understanding of TRE estimation in fiducial-based rigid-body registration, and strengthen the need for realistic and reliable FLE models and effective TRE estimation methods.