Given a finite, simple graph $G$, the $k$-component order edge connectivity of $G$ is the minimum number of edges whose removal results in a subgraph for which every component has order at most $k-1$. In general, determining the $k$-component order edge connectivity of a graph is NP-hard. We determine conditions on the vertex degrees of $G$ that can be used to imply a lower bound on the $k$-component order edge connectivity of $G$. We will discuss the process for generating such conditions for a lower bound of 1 or 2, and we explore how the complexity increases when the desired lower bound is 3 or more. In the process, we prove some related results about integer partitions.