  We present ten young ($\lesssim$10 Myr) late-K and M dwarf stars observed in
K2 Campaign 2 that host protoplanetary disks and exhibit quasi-periodic or
aperiodic dimming events. Their optical light curves show $\sim$10-20 dips in
flux over the 80-day observing campaign with durations of $\sim$0.5-2 days and
depths of up to $\sim$40%. These stars are all members of the $\rho$ Ophiuchus
($\sim$1 Myr) or Upper Scorpius ($\sim$10 Myr) star-forming regions. To
investigate the nature of these "dippers" we obtained: optical and
near-infrared spectra to determine stellar properties and identify accretion
signatures; adaptive optics imaging to search for close companions that could
cause optical variations and/or influence disk evolution; and
millimeter-wavelength observations to constrain disk dust and gas masses. The
spectra reveal Li I absorption and H$\alpha$ emission consistent with stellar
youth (<50 Myr), but also accretion rates spanning those of classical and
weak-line T Tauri stars. Infrared excesses are consistent with protoplanetary
disks extending to within $\sim$10 stellar radii in most cases; however, the
sub-mm observations imply disk masses that are an order of magnitude below
those of typical protoplanetary disks. We find a positive correlation between
dip depth and WISE-2 excess, which we interpret as evidence that the dipper
phenomenon is related to occulting structures in the inner disk, although this
is difficult to reconcile with the weakly accreting aperiodic dippers. We
consider three mechanisms to explain the dipper phenomenon: inner disk warps
near the co-rotation radius related to accretion; vortices at the inner disk
edge produced by the Rossby Wave Instability; and clumps of circumstellar
material related to planetesimal formation.
