We report the discovery and the analysis of the short timescale binary-lens microlensing event, MOA-2015-BLG-337. The lens system could be a planetary system with a very low mass host, around the brown dwarf/planetary mass boundary, or a brown dwarf binary. We found two competing models that explain the observed light curves with companion/host mass ratios of q ∼ 0.01 and ∼ 0.17, respectively. A significant finite source effect in the best-fit planetary model (q ∼ 0.01) reveals a small angular Einstein radius of θ E ≃ 0.03 mas which favors a low mass lens. We obtain the posterior probability distribution of the lens properties from a Bayesian analysis. The results for the planetary models strongly depend on a power-law index in planetary mass regime, α pl , in the assumed mass function. In summary, there are two solutions of the lens system: (1) a brown dwarf/planetary mass boundary object orbited by a super-Neptune (the planetary model with α pl = 0.49) and (2) a brown dwarf binary (the binary model). If the planetary models are correct, this system can be one of a new class of planetary system, having a low host mass and also a planetary mass ratio (q < 0.03) between the companion and its host. The discovery of the event is important for the study of planetary formation in very low mass objects. In addition, it is important to consider all viable solutions in these kinds of ambiguous events in order for the future comprehensive statistical analyses of planetary/binary microlensing events.