  We show that the spectrum of an ensemble of two-level systems can be
broadened through `resetting' discrete fluctuations, in contrast to the
well-known motional-narrowing effect. We establish that the condition for the
onset of motional broadening is that the ensemble frequency distribution has
heavy tails with a diverging first moment. We find that the asymptotic
motional-broadened lineshape is a Lorentzian, and derive an expression for its
width. We explain why motional broadening persists up to some fluctuation rate,
even when there is a physical upper cutoff to the frequency distribution.
