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.

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.

Positive results in the general setting are nowhere near as rich as when the set of nondegenrate elements is countable. The best of them is Edwards's Cell-kike Approximation Theorem, which promises that for $n>4$ a cell-like decomposition of an $n$-manifold $M$ is shrinkable if the quotient space is finite dimensional and satisfies the Disjoint Disks Property.