SEM Confidence Intervals and Repeated Measures

I'm analysing data from a repeated measures study. All of my stats tests (ANOVA and non-parametric tests where appropriate) are telling me that the groups differ but when I plot my graphs my SEM CIs are overlapping. I've always thought this meant the groups did not differ? Do SEM CIs represent something different in within subject designs?

When you say the groups differ, what do they differ on? When you say the SEM CI's overlap, which parameters are you looking at?
The differ on group means (as tested by the ANOVA) and medians (for non-parametic tests). When I plot the graphs on both excel and SPSS, I assume that I'm just plotting the SEM CIs of the groups, as attained from each individuals mean.