An interesting question. Of course there is some ambiguity in the formulation "when can we tell". 

Certainly in explicit examples, one may be able to apply ad-hoc methods. For example, things are easier for the Sierpinski gasket and carpet, since these have identifiable features in terms of their complementary regions.

For example, if we wish to write the Sierpinski gasket as a union of smaller copies, it should be fairly easy to see that each complementary region must be mapped to another complementary region. But this means that each contraction must correspond to one of the "smaller triangles" that appear in the usual gasket contraction, and we need at least three of these to make up the whole gasket.

The same type of argument should work for the Sierpinski carpet.