  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 $R\to \infty$ 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.
