Cognitive plasticity is a topic of interest since it allows us to analyse the potential cognitive modifiability of a person. Previous research has demonstrated the existence of plasticity in old age [Baltes, P. B. (1987). Theoretical propositions of life-span developmental psychology: On the dynamics between growth and decline. Developmental Psychology, 23(5), 611–626] regardless of presence or absence of cognitive deterioration [Calero, M. D., & Navarro, E. (2004). Relationship between plasticity, mild cognitive impairment and cognitive decline. Archives of Clinical Neuropsychology, 19, 653–660]. In this context, the present study was designed to analyse the presence of plasticity in elderly persons who seemed to present cognitive deterioration, and to explore the relation between cognitive plasticity and the results obtained from a memory training programme. One hundred and thirty-three elderly persons participated in the study and were evaluated by means of a cognitive plasticity test (Position test) and various tests for measuring the effects of the training. Part of the elderly population received the memory training, whose effects were measured immediately after the training and again after 9 months. The results demonstrate that the programme significantly improves cognitive performance, while plasticity is shown to be an important modulating variable on the improvement achieved.