The aims of this study were to (1) determine if older people at their fast walking speed (FWS) are able to reach the speed required at pedestrian crossings (>1.2 m/s) and (2) determine the role of cognitive impairment on the ability to alter speed and walk quickly. Participants were recruited from the Angers Memory Clinic, France. Gait speed was assessed at preferred and FWS using a GAITRite walkway. Walking speed reserve (WSR) was calculated as the difference between FWS and preferred speeds. Participants were classified into cognitive stages (cognitively healthy, mild cognitive impairment, mild and moderate dementia) based on neuropsychological evaluations. The proportion of participants with a FWS of <1.2 m/s was reported. The association between cognitive stage and preferred, fast and walking speed reserve was assessed using multivariable regression, adjusting for covariates. The mean age of the sample (n = 681) was 73.3 (SD 5.8) years. At preferred speed 73.7%, and at FWS 12.8%, of participants had speeds less than 1.2 m/s. Poorer cognitive stage was associated with slower preferred speed (β -0.08, 95% CI -0.10, -0.06), FWS (β -0.13, 95% CI -0.16, -0.10) and also with smaller WSR (m/s) (β -0.05, 95% CI -0.07, -0.03), but not WSR (%) (β -1.73, 95% CI -4.38, 0.93). In older people, worse stages of cognitive impairment were associated with poorer ability to increase speed and walk quickly. Such limitations may result in reduced ability to access the community.