The Sausage Catastrophe (Jörg Wills) is the observation that in d = 3 and d = 4, the densest packing of n spheres is a sausage for small n and jumps to a full-dimensional packing for large n without passing through any intermediate dimensions. We denote the smallest value of n for which the densest packing is full-dimensional by k_d. We discuss some upper and lower bounds for k_3 and k_4, including k_3 ≤ 56 by Wills (1985) and k_4 < 375,769 by Gandini and Zucco (1992). We present some initial improvements to the upper bound for k_4 via extending the work of Gandini and Zucco.