We report the discovery by the Palomar Transient Factory (PTF) of the transient source PTF11agg, which is distinguished by three primary characteristics: (1) bright, rapidly fading optical transient emission; (2) a faint, blue quiescent optical counterpart; and (3) an associated year-long, scintillating radio transient. We argue that these observed properties are inconsistent with any known class of Galactic transients, and instead suggest a cosmological origin. The detection of incoherent radio emission at such distances implies a large emitting region, from which we infer the presence of relativistic ejecta. The observed properties are all consistent with the population of long-duration gamma-ray bursts (GRBs), marking the first time such an outburst has been discovered in the distant universe independent of a high-energy trigger. We searched for possible high-energy counterparts to PTF11agg, but found no evidence for associated prompt emission. We therefore consider three possible scenarios to account for a GRB-like afterglow without a high-energy counterpart: an "untriggered" GRB (lack of satellite coverage), an "orphan" afterglow (viewing-angle effects), and a "dirty fireball" (suppressed high-energy emission). The observed optical and radio light curves appear inconsistent with even the most basic predictions for off-axis afterglow models. The simplest explanation, then, is that PTF11agg is a normal, on-axis long-duration GRB for which the associated high-energy emission was simply missed. However, we have calculated the likelihood of such a serendipitous discovery by PTF and find that it is quite small (~ 2.6%). While not definitive, we nonetheless speculate that PTF11agg may represent a new, more common (> 4 times the on-axis GRB rate at 90% confidence) class of relativistic outbursts lacking associated high-energy emission. ; Comment: 45 pages, 7 figures; Resubmitted to ApJ after incorporating referee comments