Here we offer an approach for being reasonably sure that finite element determinations of stress concentration factors are accurate enough to be included in engineering handbooks. The approach has two contributors. The first consists of analyzing a stress concentration on a sequence of systematically refined meshes until the error estimates of ASME have that sufficient accuracy has been achieved. The second consists of constructing a test problem with an exact and somewhat higher value of its stress concentration factor, then analyzing this test problem with the same sequence of meshes and showing that, in fact, sufficient accuracy has been achieved. In combination, these two means of verification are applied to a series of U-notches in a plate under tension. Together they show that it is reasonable to regard finite element values of stress concentration factors on the finest meshes as being accurate to three significant figures. Given this level of accuracy, it is then also reasonable to use the approach to verify other existing stress concentration factors and resolve any discrepancies between them, as well as to verify new stress concentration factors.