The standard construction of the minimal-order state observer for a linear time-invariant plant is shown to be a degenerate form of the ordinary full-order observer. The naive state estimation technique of successively differentiating the plant output is also shown to be a limiting case of the same full-order structure. Various results concerning the set of all degenerate forms of the full-order observer are presented. In particular, conditions are established under which shifting observer eigenvalues far to the left in the complex plane implies proximity to a successively differentiating system. In addition, it is shown that such a scheme can never yield satisfactory estimation. Singular perturbation and singular system theory are used throughout the analysis.