We combine new Wide Field Camera 3 IR Channel (WFC3/IR) F160W (H-160) imaging data for NGC 1399, the central galaxy in the Fornax cluster, with archival F475W (g(475)), F606W (V-606), F814W (I-814), and F850LP (z.(850)) optical data from the Advanced Camera for Surveys (ACS). The purely optical g(475)-I-814, V-606-I-814, and g(475)-z(850) colors of NGC 1399's rich globular cluster (GC) system exhibit clear bimodality, at least for magnitudes I-814 > 21.5. The optical-IR I-814-H-160 color distribution appears unimodal, and this impression is confirmed by mixture modeling analysis. The V-606-H-160 colors show marginal evidence for bimodality, consistent with bimodality in V-606-I-814 and unimodality in I-814-H-160. If bimodality is imposed for I-814-H-160 with a double Gaussian model, the preferred blue/red split differs from that for optical colors; these "differing bimodalities" mean that the optical and optical-IR colors cannot both be linearly proportional to metallicity. Consistent with the differing color distributions, the dependence of I-814-H-160 on g(475)-I-814 for the matched GC sample is significantly nonlinear, with an inflection point near the trough in the g(475)-I-814 color distribution; the result is similar for the I-814-H-160 dependence on g(475)-z(850) colors taken from the ACS Fornax Cluster Survey. These g(475)-z(850) colors have been calibrated empirically against metallicity; applying this calibration yields a continuous, skewed, but single-peaked metallicity distribution. Taken together, these results indicate that nonlinear color-metallicity relations play an important role in shaping the observed bimodal distributions of optical colors in extragalactic GC systems.