The motion of bodies through fluid at low Reynolds number is appreciably affected by the container walls. Consequently the Stokes-flow theory due to Batchelor (1970) and others for a slender body falling in an unbounded fluid is difficult to test experimentally unless it is extended to take account of nearby boundaries. Theoretical expressions are given here for certain drag coefficients of a circular cylindrical slender rod of finite length falling close to a single plane wall or falling midway between two parallel plane walls. Experiments with a very viscous liquid are described in which cylinders of small thickness-to-length ratios (ranging from 1:10 to 1:100 approximately) are made to fall in suitable orientations. From their times of fall over a measured distance experimental drag coefficients are determined and compared with the corresponding theoretical value from the extension of Batchelor's theory. For rods falling in a horizontal orientation the theoretical and experimental results are consistent within the order of accuracy of the experiments. However, when results are compared for rods falling in a vertical orientation there is a significant difference for which possible explanations are presented.