We present an analysis of the active galaxy SDSS J131642.90+175332.5, which is remarkable because all of its narrow emission lines are double-peaked, and because it additionally shows an extra broad component (FHWM ∼ 1400 km s −1 ) in most of its forbidden lines, peaking in between the two narrow systems. The peaks of the two narrow systems are separated by 400-500 km s −1 in velocity space. The spectral characteristics of double-peaked [O iii] emission have previously been interpreted as a signature of dual or binary active galactic nuclei (AGNs), among other models. In the context of the binary scenario, SDSS J131642.90+175332.5 is a particularly good candidate because not just one line but all of its emission lines are double-peaked. However, we also discuss a number of other scenarios which can potentially account for double-peaked narrow emission lines, including projection effects, a two-sided outflow, jet-cloud interactions, special narrow-line region (NLR) geometries (disks, bars, or inner spirals), and a galaxy merger with only one AGN illuminating two NLRs. We argue that the similarity of the emission-line ratios in both systems, and the presence of the very unusual broad component at intermediate velocity, makes a close pair of unrelated AGNs unlikely, and rather argues for processes in a single galaxy or merger. We describe future observations that can distinguish between these remaining possibilities.