Given a set of experimentally determined lower and upper bounds on the distances between the atoms of a molecule, we study the minimum and maximum values that any one distance can attain when all of the remaining distances are confined between their lower and upper bounds. The triangle inequality may be used to derive a first approximation to these `distance limits', and a complete characterization of these `triangle limits', together with an efficient algorithm for computing them, is presented. A four-point relation known as the `tetrangle inequality' is then discussed as a possible means of obtaining an improved approximation.