The pathological effects of multiple sclerosis are not confined to lesions; tissues that appear normal on conventional magnetic resonance imaging scans are also affected, albeit subtly. One imaging technique that has proven sensitive to such effects is T1-relaxation time measurement, with previous work demonstrating abnormalities in normal-appearing white matter and grey matter. In this work we investigated the evolution of T1-relaxation time changes in normal-appearing white matter and grey matter in relapsing-remitting multiple sclerosis. Three- and five-year follow-up data from 35 people with clinically early (a mean of 1.6 years from first clinical event) relapsing-remitting multiple sclerosis and 15 healthy controls were analysed. T1-relaxation time histograms were extracted from normal-appearing white matter and grey matter, and mean, peak height and peak location values were estimated. T1-relaxation time peak height declined in the multiple sclerosis normal-appearing white matter and grey matter, but not the control group (rate difference p = 0.024 in normal-appearing white matter, in normal-appearing grey matter p = 0.038); other T1-relaxation time changes were not significantly different between groups. Changes in T1-relaxation time measures did not correlate with increases in brain T2-weighted lesion loads or Expanded Disability Status Scale scores. These results suggest that the processes underlying changes in normal-appearing white matter and grey matter T1-relaxation times are not immediately linked to white matter lesion formation, and may represent more diffuse but progressive sub-clinical pathology in relapsing-remitting multiple sclerosis.