© 2016. The American Astronomical Society. All rights reserved.. We present observations of four rapidly rising (trise ≈ 10 days) transients with peak luminosities between those of supernovae (SNe) and superluminous SNe (Mpeak ap; -20) - one discovered and followed by the Palomar Transient Factory (PTF) and three by the Supernova Legacy Survey. The light curves resemble those of SN 2011kl, recently shown to be associated with an ultra-long-duration gamma-ray burst (GRB), though no GRB was seen to accompany our SNe. The rapid rise to a luminous peak places these events in a unique part of SN phase space, challenging standard SN emission mechanisms. Spectra of the PTF event formally classify it as an SN II due to broad Hα emission, but an unusual absorption feature, which can be interpreted as either high velocity Hα (though deeper than in previously known cases) or Si ii (as seen in SNe Ia), is also observed. We find that existing models of white dwarf detonations, CSM interaction, shock breakout in a wind (or steeper CSM), and magnetar spin down cannot readily explain the observations. We consider the possibility that a "Type 1.5 SN" scenario could be the origin of our events. More detailed models for these kinds of transients and more constraining observations of future such events should help to better determine their nature.