We present a 16-vertex tetrahedralization of S³ (the 3-sphere) for which no topological bistellar flip other than a 1-to-4 flip (i.e., a vertex insertion) is possible. This answers a question of Altshuler et al. which asked if any two n-vertex tetrahedralizations of S³ are connected by a sequence of 2-to-3 and 3-to-2 flips. The corresponding geometric question is whether two tetrahedralizations of a finite point set S in ³ in general position are always related via a sequence of geometric 2-to-3 and 3-to-2 flips. Unfortunately, we show that this topologically unflippable complex and others with its properties cannot be geometrically realized in ³.