We report on our measurements of resistance fluctuations as a function of magnetic field B in an ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As/GaAs heterostructure of etched width w=2.5 \ensuremath{\mu}m in the integral- and fractional-quantum-Hall-effect regimes. High-frequency fluctuations are observed near the longitudinal resistance (${\mathit{R}}_{\mathit{x}\mathit{x}}$) minima for \ensuremath{\nu}=1, 2, 3, 4, and 1/3. The quasiperiods \ensuremath{\Delta}B(\ensuremath{\nu} = integer) of the fluctuations for integer \ensuremath{\nu} are all \ensuremath{\sim}0.016 T, while for \ensuremath{\nu}=1/3, the quasiperiod \ensuremath{\Delta}B(\ensuremath{\nu} = 1/3) is \ensuremath{\sim}0.05 T, or a factor of 3 larger. The fluctuations at integer \ensuremath{\nu} are consistent with inter-edge-state tunneling via magnetically bound states encircling a potential hill of a diameter roughly equal to the conducting width of the channel. A similar model, with the difference that the tunneling is by quasiparticles of fractional charge ${\mathit{e}}^{\mathrm{*}}$=e/q, predicts a scaling of the quasiperiod as \ensuremath{\Delta}B(\ensuremath{\nu}=1/q)=q \ensuremath{\Delta}B(\ensuremath{\nu} = integer).Interpreted in terms of this model, the data provide direct evidence of the existence of quasiparticles of charge ${\mathit{e}}^{\mathrm{*}}$=e/3 in the \ensuremath{\nu}=1/3 fractional quantum Hall effect. For both \ensuremath{\nu}=1/3 and \ensuremath{\nu}= integer, the individual fluctuation patterns for different pairs of voltage probes are strongly correlated only if the pairs share a length of the channel, indicating that the source of the fluctuations is local, as predicted by the model. A Coulomb blockade as the origin of the fluctuations is ruled out by the fact that for \ensuremath{\nu}=1 and 2 the fluctuation amplitudes saturate at temperatures ${\mathit{T}}_{\mathit{c}}$(\ensuremath{\nu}=1)\ensuremath{\simeq}66 mK and ${\mathit{T}}_{\mathit{c}}$(\ensuremath{\nu}=2)\ensuremath{\simeq}121 mK, and also saturate at currents ${\mathit{I}}_{\mathit{c}}$(\ensuremath{\nu}=1)\ensuremath{\simeq}0.5 nA and ${\mathit{I}}_{\mathit{c}}$(\ensuremath{\nu}=2)\ensuremath{\simeq}1.7--3.0 nA. These results indicate that for integer \ensuremath{\nu}, the bound-state-energy spacing \ensuremath{\Delta}\ensuremath{\varepsilon}(\ensuremath{\nu}) scales as \ensuremath{\nu} or ${\mathit{B}}^{\mathrm{\ensuremath{-}}1}$, inconsistent with a Coulomb blockade.