  [ABRIDGED] We report the results of a $J$ band search for cloud-related
variability in the atmospheres of 62 L4-T9 dwarfs using the Du Pont 2.5-m
telescope at Las Campanas Observatory and the Canada France Hawaii Telescope on
Mauna Kea. We find 9 of 57 objects included in our final analysis to be
significantly variable with >99% confidence, 5 of which are new discoveries. In
our study, strong sinusoidal signals (peak-to-peak amplitudes >2%) are confined
to the L/T transition (4/16 objects with L9-T3.5 spectral types and 0/41
objects for all other spectral types). The probability that the observed
occurrence rates for strong variability inside and outside the L/T transition
originate from the same underlying true occurrence rate is excluded with >99.7%
confidence. These observations suggest that the settling of condensate clouds
below the photosphere in brown dwarf atmospheres does not occur in a spatially
uniform manner. Rather, the formation and sedimentation of dust grains at the
L/T transition is coupled to atmospheric dynamics, resulting in highly
contrasting regions of thick and thin clouds and/or clearings. Outside the L/T
transition we identify 5 weak variables (peak-to-peak amplitudes of 0.6%-1.6%).
Excluding L9-T3.5 spectral types, we infer that $60^{+22}_{-18}$% of targets
vary with amplitudes of 0.5%$-$1.6%, suggesting that surface heterogeneities
are common among L and T dwarfs. Our survey establishes a significant link
between strong variability and L/T transition spectral types, providing
evidence in support of the hypothesis that cloud holes contribute to the abrupt
decline in condensate opacity and 1 micron brightening observed in this regime.
More generally, fractional cloud coverage is an important model parameter for
brown dwarfs and giant planets, especially those with L/T transition spectral
types and colors.
