Recent observations confirm the presence of large-scale magnetic field of microgauss strength over approximately 10 kpc scales in some clusters of galaxies. The possible production of these fields by a turbulent dynamo driven by galaxy motion through the ICM is examined, using a variety of seed fields. The calculations are fully non-linear and time-dependent, and they show that it is in general very difficult for this mechanism to produce the observed magnetic fields. This result differs from previous calculations which did not include the evolution over a wide range of scales, and it arises primarily from the inclusion of the cascade of turbulent energy from large scales to smaller scales in a self-consistent manner. This process should be at work in all clusters, and its inability to readily produce the observed field strengths and scales may explain why these fields are not directly observed in all clusters and may imply the need for exceptional conditions to exist in those clusters where such fields are seen.