The critical exponent $\ensuremath{\beta}$ for magnetically ordered materials has been obtained from a variety of hyperfine experiments, such as nuclear magnetic resonance, perturbed angular correlations, and M\"ossbauer effect. In this paper we discuss probe disturbance effects on hyperfine measurements of $\ensuremath{\beta}$. We consider both chemically pure substances and materials into which the hyperfine probe has been introduced as a dilute impurity, and emphasize the latter. From a theoretical point of view, we investigate several molecular-field models and present results of new calculations for an isolated nonmagnetic impurity in a three-dimensional Ising model. It is found that the disturbance produced by the impurity is substantially smaller in the latter case than for the corresponding molecular-field model. From an experimental point of view we present a survey of cases for which bulk and hyperfine measurements have been made on the same substance. We also report on a reanalysis of data in Ni, and present the results of power-law fits made for various ranges of reduced temperature. The cases studied include Ni $^{100}\mathrm{Rh}$, Ni $^{111}\mathrm{Cd}$, Ni $^{57}\mathrm{Fe}$, and Ni $^{181}\mathrm{Ta}$, and are restricted to samples produced by diffusion or melting. On the available theoretical and experimental evidence, we conclude that values of $\ensuremath{\beta}$ determined from hyperfine measurements involve probe-disturbance errors that are certainly smaller than 2% if the reduced temperature is restricted to $tl{10}^{\ensuremath{-}2}$ and care is taken to avoid source inhomogeneities and temperature gradients.