Bing's Dogbone Space stems from a decomposition of  $S^3$  unto points
and an uncountable collection of tame arcs.  The decomposition space is not a manifold,
so the decomposition is non-shrinkable.  W. T. Eaton (Proc. Amer. Math. Soc. 39
(1973), 379--387) presented a higher dimensional analog of the Dogbone Space, involving a decomposition of $S^n$, $n>3$, into points and tame arcs that yields a nonmanifold.