There is a popular myth that Galileo dropped two objects of the same shape but different mass, noted their equal fall time, and concluded that gravitational motion is independent of the mass of the object. This paper demonstrates that this experiment--if actually performed--most likely would have yielded a different result and thus with modern eyes led to a different conclusion. The paper consists of two parts: (1) a theoretical description with a numerical and an analytical solution of the problem, and (2) an experiment of the Galilean type that provides experimental evidence within 1% accuracy that the theory applies. Previous papers on this subject have been almost exclusively theoretical and/or calculational, and we emphasize the suitability of the performed experiment as an affordable instructional exercise, e.g. for undergraduates. We find a difference in impact time of 0.109 s for otherwise nearly identical objects of masses differing by about a factor of 3, m[subscript light] = 57.52 g and m[subscript heavy] = 173.62 g, dropped from a height of 23.192 m, about half of the height of the Leaning Tower of Pisa. Very convincing agreement between experiment and theory, including an account of the fluid dynamics in air, is achieved. By use of the theory thus corroborated, the calculated difference in height at first impact corresponding to the Galilean case is about 5.8 m (assuming a lead ball and an ebony ball, each the size of a tennis ball), easily detectable both visually and, with a difference in impact time of about 0.22 s, aurally. The time difference is also sufficiently large to be unlikely to be caused by inaccuracy in release time. This gives an indication that if Galileo did actually perform the experiment, he may have chosen to neglect small differences (?10%) in impact times observed using large differences (factor ?10) in mass.