We have measured a paramagnetic Meissner effect in Nb-Al2O3-Nb Josephson-junction arrays using a scanning superconducting quantum interference device microscope. The arrays exhibit diamagnetism for some cooling fields and paramagnetism for other cooling fields. The measured mean magnetization is always small, < Phi (tot) - Phi (ext)> < 0.3(0) tin terms of flux per unit cell of the array, where Phi (0) is the flux quantum), for the range of cooling fields investigated (- 12 Phi (0) to 12 Phi (0)). We demonstrate that a different model of magnetic screening, valid for multiply-connected superconductors, reproduces all of the essential features of paramagnetism that we observe and that no exotic mechanism, such as d-wave superconductivity, is needed for paramagnetism.