We report the discovery of a highly eccentric, double-lined spectroscopic binary star system (TYC 3010-1494-1), comprising two solar-type stars that we had initially identified as a single star with a brown dwarf companion. At the moderate resolving power of the MARVELS spectrograph and the spectrographs used for subsequent radial-velocity (RV) measurements (R <~ 30, 000), this particular stellar binary mimics a single-lined binary with an RV signal that would be induced by a brown dwarf companion (Msin i ~ 50 M Jup) to a solar-type primary. At least three properties of this system allow it to masquerade as a single star with a very-low-mass companion: its large eccentricity (e ~ 0.8), its relatively long period (P ~ 238 days), and the approximately perpendicular orientation of the semi-major axis with respect to the line of sight (ω ~ 189°). As a result of these properties, for ~95% of the orbit the two sets of stellar spectral lines are completely blended, and the RV measurements based on centroiding on the apparently single-lined spectrum is very well fit by an orbit solution indicative of a brown dwarf companion on a more circular orbit (e ~ 0.3). Only during the ~5% of the orbit near periastron passage does the true, double-lined nature and large RV amplitude of ~15 km s–1 reveal itself. The discovery of this binary system is an important lesson for RV surveys searching for substellar companions; at a given resolution and observing cadence, a survey will be susceptible to these kinds of astrophysical false positives for a range of orbital parameters. Finally, for surveys like MARVELS that lack the resolution for a useful line bisector analysis, it is imperative to monitor the peak of the cross-correlation function for suspicious changes in width or shape, so that such false positives can be flagged during the candidate vetting process.