  The peculiar nebular emission displayed by galaxies in the early Universe
presents a unique opportunity to gain insight into the regulation of star
formation in extreme environments. We investigate 500 (109) galaxies with deep
NIRSpec/PRISM observations from the JADES survey at $z>2$ ($z>5.3$), finding 52
(26) galaxies with Balmer line ratios more than $1\sigma$ inconsistent with
Case B recombination. These anomalous Balmer emitters (ABEs) cannot be
explained by dust attenuation, indicating a departure from Case B
recombination. To address this discrepancy, we model density-bounded nebulae
with the photoionisation code CLOUDY. Density-bounded nebulae show anomalous
Balmer line ratios due to Lyman line pumping and a transition from the nebulae
being optically thin to optically thick for Lyman lines with increasing cloud
depth. The H$\alpha$/H$\beta$ versus H$\gamma$/H$\beta$ trend of
density-bounded models is robust to changes in stellar age of the ionising
source, gas density, and ionisation parameter; however, increasing the stellar
metallicity drives a turnover in the trend. This is due to stronger stellar
absorption features around Ly$\gamma$ reducing H$\beta$ fluorescence, allowing
density-bounded models to account for all observed Balmer line ratios. ABEs
show higher [OIII]/[OII], have steeper ultra-violet slopes, are fainter, and
are more preferentially Ly$\alpha$ emitters than galaxies which are consistent
with Case B and little dust. These findings suggest that ABEs are galaxies that
have become density bounded during extreme quenching events, representing a
transient phase of $\sim$20 Myr during a fast breathing mode of star formation.
