We study a quantum particle coupled to hard-core bosons and propagating on disordered ladders with $R$ legs. The particle dynamics is studied with the help of rate equations for the boson-assisted transitions between the Anderson states. We demonstrate that for finite $R < infty$ and sufficiently strong disorder the dynamics is subdiffusive, while the two-dimensional planar systems with $Rto infty$ appear to be diffusive for arbitrarily strong disorder. The transition from diffusive to subdiffusive regimes may be identified via statistical fluctuations of resistivity. The corresponding distribution function in the diffusive regime has fat tails which decrease with the system size $L$ much slower than $1/sqrt{L}$. Finally, we present evidence that similar non--Gaussian fluctuations arise also in standard models of many-body localization, i.e., in strongly disordered quantum spin chains.
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