We present an analysis of more than 3700 observations of the Ca II H and K lines in 57 Sun-like stars and over 3000 analogous observations of the Sun. Ten of the 57 stars under consideration are observed in flat states, but these stars do not always exhibit overall Ca II H and K core brightness below that of solar minimum. Solar activity minimum lies near the lowest level observed for stars with cyclic or irregular variability, but many flat stars have HK activity levels comparable to or exceeding that of solar minimum. While flat activity stars may be in periods of extended activity minima analogous to the solar Maunder minimum, a significant reduction in magnetic activity during such periods is not implied (although it is also not rejected) by the data.