The Yale/SMARTS optical-near-IR monitoring program has followed the variations in emission of the Fermi-LAT monitored blazars in the southern sky with closely spaced observations since 2008. We report the discovery of an optical-near-IR (OIR) outburst with no accompanying gamma-rays in the blazar PKS 0208-512, one of the targets of this program. While the source undergoes three outbursts of 1 mag or more at OIR wavelengths lasting for longer than 3 months during 2008-2011, only interval 1 and 3 have corresponding bright phases in GeV energies lasting longer than 1 month. The OIR outburst during interval 2 is comparable in brightness and temporal extent to the OIR flares during intervals 1 and 3 which do have gamma-ray counterparts. Gamma-ray and OIR variability are very well-correlated in most cases in the Fermi blazars and the lack of correlation in this case is anomalous. By analyzing the gamma-ray, OIR, and supporting multi-wavelength variability data in details, we speculate that the location of the outburst in the jet during interval 2 was closer to the black hole where the jet is more compact and the magnetic field strength is higher, and the bulk Lorentz factor of the material in the jet is smaller. These result in a much lower Compton dominance and no observable gamma-ray outburst during interval 2.