This article demonstrates that increasing the number of measurements made on a system beyond the minimum (i.e., the number of degrees of freedom in the system) can reduce the effect of measurement errors. The article shows, for three broad classes of measurement problems, that, as the measurement redundancy increases, the residual error falls to a small constant value. Twice the number of degrees of freedom will allow performance very close to the ultimate limits. The article presents one example within each class; however, a vast number of other measurement problems have a formulation identical to one of these three. The first example is the calibration of a resistive voltage divider consisting of a number of nominally equal resistors in series. The second is the determination of the complex response of a filter whose output can be observed only through a power detector. The third is the calibration of a resistive current combiner or current mode digital-to-analog converter (DAC). The basic ideas in the article are general and extensions to other measurement problems outside of the three categories should be straightforward, although the details of the solution will be problem dependent.