If you consider upper semicontinuous decompositions on compact connected sets, then, in this paperthis paper it is proved that it is not possible to fill any euclidean space in such a way.
There is a paper by Robertspaper by Roberts where he proves the two dimensional result and also gives an example of an upper semicontinuous decomposition of the plane into cellular curves (they may be not simple, but they are a decreasing intersection of disks).
I know this does not respond the question entirely, since the upper semicontinuous hypothesis is strong, however, many times desirable.
