In practical propagation environments, different massive MIMO users can have correlated angles in spatial paths. In this paper, we study the effect of angle correlation on inter-user channel correlation via a combination of measurement and analysis. We show three key results. First, we collect a massive MIMO channel dataset for examining the inter-user channel correlation in a real-world propagation environment; the dataset is now open-access. We observed channel correlation higher than 0.48 for all close-by users. Additionally, over 30% of far-away users, even when they are tens of wavelengths apart, have inter-user channel correlation that is at least twice higher than the correlation in the i.i.d. Rayleigh fading channel. Second, we compute the inter-user channel correlation in closed-form as a function of inter-user angle correlation, the number of base-station antennas, and base-station inter-antenna spacing. Our analysis shows that inter-user angle correlation increases the inter-user channel correlation. Inter-user channel correlation reduces with a larger base-station array aperture, i.e., more antennas and larger inter-antenna spacing. Third, we explain the measurements with numerical experiments to show that inter-user angle correlation can result in significant inter-user channel correlation in practical massive MIMO channels.