"These tests of attention require intact short-term memory. Most patients with brain disorders have intact short-term memory when recall follows repetition immediately, with neither delay nor interference. When their attention is directed away from tasks by an interpolated activity, retention, even for relatively brief intervals, becomes tenuous. Patients with actual short-term memory impairment do exist but it is relatively rare (e.g. Vallar & Shallice, 1990; Warrington & Shallice, 1984), so this possibility should be evaluated. Poor performance on a simple digit span task is more likely to be representative of an attentional impairment rather than a true memory impairment.
Attentional capacity is resistant to the effects of many brain disorders. It may be restricted in the first months following head trauma but it is likely to return to normal during latter states (Bazarian et al., 1999; Lezak, 1979; Ponsford & Kinsells, 1992). Most mildly demented Alzheimer's patients have normal capacity for reciting a string of digits (Pachana et al., 1996; Rubin et al., 1992). However, when the information becomes more complex, as in sentence span tests, or more information is presented than can normally be grasped at once, as in supraspan tests (Benton et al., 1983; Milner, 1970; see also Lezak, 1995), the reduced attentional capacity of many brain-injured persons becomes evident."
- From The Handbook of Memory Disorders, page 639
- See also this link for some information regarding the concept of attention, as it relates to the issues noted above.
Basically, what this passage indicates is that organic brain issues typically do not impact short-term memory; to the extent an individual is displaying problems with short-term memory, the actual issue is most likely related to attention.