We study two different but closely related vulnerability measures of graphs, namely vertex integrity and component order connectivity. Both parameters take into account not only the number of vertices that need to be deleted in order to break a graph into pieces, but also the number of vertices in the largest connected component that remains. Kratsch et al. (Discr. Appl. Math. 77: 259-270, 1997) presented polynomial-time algorithms for computing the vertex integrity for various classes of intersection graphs. In particular, they showed that this problem can be solved in O(n^{2d+1}) time on co-comparability graphs of dimension at most d, provided that an intersection model of the input graph is given as part of the input. The weighted version of the problem is known to be solvable in polynomial time on interval graphs, but is NP-complete on chordal graphs. We present several new classical and parameterized complexity results for the weighted and unweighted versions of the problem. In particular, we show that determining the (unweighted) vertex integrity is NP-complete on co-comparability graphs, thereby refuting a conjecture by Ray and Deogun (J. Combin. Math. Combin. Comput. 16: 65-73, 1994). We also obtain several classical and parameterized complexity results involving the component order connectivity, uncovering interesting similarities and differences between the two vulnerability measures.