Statistical analysis of relatively large samples of Faraday rotation measurements of radio sources seen through clusters of galaxies recently led Kim, Tribble, & Kronberg to conclude that magnetic fields of microgauss strength were present in clusters of galaxies. Expanding the analysis, we have checked these samples for the possible effects of Galactic contamination, of a selection bias due to inclusion of cluster sources into the test sample, and of cooling flows. We find no statistically significant evidence that the seemingly loose selection criteria affected the results. Through comparisons of the likely spatial distributions of the fields and relativistic electrons, detailed relations are obtained between field values derived from Faraday rotation, and combined radio and high-energy X measurements. The observational results are compared with predictions of the turbulent dynamo theory of magnetic field generation in clusters. It is shown that in order to account for the observed rotation measures, turbulent dynamo models, which predict ''thin rope'' morphology for the fields, require prohibitively high field values. This and other considerations lead to the conclusion that current turbulent dynamo models cannot viably account for the observed properties of intracluster fields. It seems more probable that the fields originate in the dispersed, magnetized interstellar media of the cluster galaxies.