Background: Computer-based programs have been implemented from a psychosocial approach for the care of people with dementia (PwD). However, several factors may determine adherence of older PwD to this type of treatment. The aim of this paper was to identify the sociodemographic, cognitive, psychological, and physical-health determinants that helped predict adherence or not to a “GRADIOR” computerized cognitive training (CCT) program in people with mild cognitive impairment (MCI) and mild dementia. Method: This study was part of a randomized clinical trial (RCT) (ISRCTN: 15742788). However, this study will only focus on the experimental group (n = 43) included in the RCT. This group was divided into adherent people (compliance: ≥60% of the sessions and persistence in treatment up to 4 months) and non-adherent. The participants were 60–90 age and diagnosed with MCI and mild dementia. We selected from the evaluation protocol for the RCT, tests that evaluated cognitive aspects (memory and executive functioning), psychological and physical health. The CCT with GRADIOR consisted of attending 2–3 weekly sessions for 4 months with a duration of 30 min Data analysis: Phi and Biserial-point correlations, a multiple logical regression analysis was obtained to find the adherence model and U Mann–Whitney was used. Results: The adherence model was made up of the Digit Symbol and Arithmetic of Wechsler Adult Intelligence Scale (WAIS-III) and Lexical Verbal Fluency (LVF) -R tests. This model had 90% sensitivity, 50% specificity and 75% precision. The goodness-of-fit p-value of the model was 0.02. Conclusions: good executive functioning in attention, working memory (WM), phonological verbal fluency and cognitive flexibility predicted a greater probability that a person would be adherent.