It is well known that the Hausdorff dimension of the invariant set $\Lambda_t$ of an iterated function system ${\mathcal F}_t$ on $\mathbb{R}^n$ depending smoothly on a parameter $t$ does not vary continuously. In fact, it has been shown recently that in general it varies lower-semi-continuously. For a specific family of systems we investigate numerically the conjecture that discontinuities in the dimension only arise when in some iterate of the iterated function system two (or more) of its branches coincide. This happens in a set of co-dimension one, but which is dense. All the other points are conjectured to be points of continuity.