This note demonstrates that measurement invariance does not guarantee meaningful and valid group comparisons in multiple-population settings. The article follows on a recent critical discussion by Robitzsch and Lüdtke, who argued that measurement invariance was not a pre-requisite for such comparisons. Within the framework of common factor analysis, we show that measurement invariance is in general not a sufficient condition for evaluating comparable constructs in all groups under consideration. We conclude with a discussion of related issues pertaining to population invariant measurement and the examination of latent differences and similarities in multi-group studies.