An illustrative example tube handling machine includes a lifter configured to receive a plurality of tubes when at least a portion of the lifter is at a first position. The lifter is configured to lift the plurality of tubes to a second, higher vertical position. A plurality of cells are respectively configured for at least temporarily holding the plurality of tubes. A first one of the cells is situated to receive the plurality of tubes from the lifter at the second vertical position. The cells are situated relative to each other such that the plurality of tubes can be sequentially transferred among the cells from the first one of the cells to a last one of the cells that releases the plurality of tubes to a delivery location.