We present multiple epochs of H-alpha spectroscopy for 47 members of the open cluster NGC 3766 to investigate the long term variability of its Be stars. Sixteen of the stars in this sample are Be stars, including one new discovery. Of these, we observe an unprecedented 11 Be stars that undergo disk appearances and/or near disappearances in our H-alpha spectra, making this the most variable population of Be stars known to date. NGC 3766 is therefore an excellent location to study the formation mechanism of Be star disks. From blue optical spectra of 38 cluster members and existing Stromgren photometry of the cluster, we also measure rotational velocities, effective temperatures, and polar surface gravities to investigate the physical and evolutionary factors that may contribute to the Be phenomenon. Our analysis also provides improvements to the reddening and distance of NGC 3766, and we find E(B-V) = 0.22 +/- 0.03 and (V-M_V)_0 = 11.6 +/- 0.2, respectively. The Be stars are not associated with a particular stage of main-sequence evolution, but they are a population of rapidly rotating stars with a velocity distribution generally consistent with rotation at 70-80% of the critical velocity, although systematic effects probably underestimate the true rotational velocities so that the rotation is much closer to critical. Our measurements of the changing disk sizes are consistent with the idea that transitory, nonradial pulsations contribute to the formation of these highly variable disks.
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