Globular cluster systems of 17 elliptical galaxies have been studied in the Coma Cluster of galaxies. Surface-brightness fluctuations have been used to determine total populations of globular clusters, and specific frequency (SN) has been evaluated for each individual galaxy. Enormous differences in SN between similar galaxies are found. In particular, SN results vary by an order of magnitude from galaxy to galaxy. Extreme cases are the following: (1) at the lower end of the range, NGC 4673 has SN=1.0+/-0.4, a surprising value for an elliptical galaxy, but typical for spiral and irregular galaxies; (2) at the upper extreme, MCG +5-31-063 has SN=13.0+/-4.2 and IC 4051 has SN=12.7+/-3.2, and they are more likely to belong to supergiant cD galaxies than to ``normal'' elliptical galaxies. Furthermore, NGC 4874, the central supergiant cD galaxy of the Coma Cluster, also exhibits a relatively high specific frequency (SN=9.0+/-2.2). The other galaxies studied have SN in the range [2, 7], the mean value being SN=5.1. No single scenario seems to account for the observed specific frequencies, so the history of each galaxy must be deduced individually by suitably combining the different models (in situ, mergers, and accretions). The possibility that Coma is formed by several subgroups is also considered. If only the galaxies of the main subgroup defined by Gurzadyan & Mazure are used, a trend in SN arises in the sense of SN being bigger in higher-density regions. This result needs further confirmation.