Are there any circumstances under which an accurate measurement may not be precise?
As the question suggests, accuracy and precision are two different things. In the current context, accuracy refers to how close a measurement comes to the known actual dimensions, whether measured in volume, length, or weight. Precision, on the other hand, is the result of repeated trials or measurements with the same results. In other words, multiple measurements of the length of a board that all come out the same is an example of precision. One knows precisely what is the length of the board. A measurement that is conducted only once may be accurate in terms of proximity to actual length, but it is not necessarily precise. Whereas accuracy can be obtained with reasonably precise tools, precision can only be attained with high-performance tools properly calibrated. Measurements that mirror each other are not precise if the tools or equations are flawed. It is easy to repeat the same mistake if one is not aware of an underlying flaw or defect in one's instruments of measurement or in one's equations.
Scientists have labored to determine the circumference of the Earth since Erasthenes (c. 276 BC-c. 195 BC), an ancient Greek studying in ancient Egypt. Whereas Erasthenes used the length of stadiums as his tool of measurement and the phenomenon of parallax involving the position of the sun relative to points on the ground, and succeeded in producing a reasonably accurate estimate of the Earth's circumference, his system of measurement clearly lacked precision. Today, man-made satellites orbit the Earth making precise measurements using the same basic concept as Erasthenes, but with considerably greater precision given the technological sophistication of their instrumentation and their controllers' ability to repeat the exercises, which involve two satellites looking at the same spot on Earth from different positions in space.