  We investigate higher-order breathers of the cubic nonlinear Schr\"odinger
equation on an elliptic background. We find that, beyond first-order, any
arbitrarily constructed breather is a single-peaked solitary wave on a
disordered background. These "quasi-rogue waves" are also common on periodic
backgrounds. We assume the higher-order breather is constructed out of
constituent first-order breathers with commensurate periods (i.e., higher-order
harmonic waves). In that case, one obtains "quasi-periodic" breathers with
distorted side-peaks. Fully periodic breathers are obtained when their
wavenumbers are harmonic multiples of the background and each other. They are
truly rare, requiring finely-tuned parameters. Thus, on a periodic background,
we arrive at the paradoxical conclusion that the apparent higher-order rogue
waves are rather common, while the truly periodic breathers are exceedingly
rare.
