The adjustable robust counterpart (ARC) of an uncertain linear program extends the robust counterpart (RC) by allowing some decision variables to adjust to the realizations of some uncertain parameters. The ARC may produce a less conservative and costly solution than the RC does but cases are known in which it does not. While the literature documents some examples of cost savings provided by adjustability (particularly affine adjustability), it is not straightforward to determine in advance whether they will materialize. The affine adjustable robust counterpart, while having a tractable structure, still may be much larger than the original problem. We establish conditions under which affine adjustability may lower the optimal cost with a numerical condition that can be checked in small representative instances. As demonstrated in applications, the conditions provide insights into constraint relationships that allow adjustability to have its intended effect.