Select the Chamfer along smoothly joined edges option in the Chamfer Vertices dialog box if you want to chamfer connected edges that continue smoothly, without abrupt change in tangency, from the selected edges.

In the example below, the location indicated with a point on the left solid was used to create the chamfer on the right. Because the Chamfer along smoothly joined edges option was enabled, the entire front portion of the object was included in the operation. If Chamfer along smoothly joined edges had not been checked, the operation would have failed while trying to chamfer the rounded portion by itself (error: "could not find acceptable sequence of capping faces to trim blend face").