An ill-conditioned linear system can give wildly wrong answers even when the algorithm is flawless and the arithmetic exact to 16 digits. Why is that a property of the problem rather than a failure of the method?

D. Because conditioning measures how much the solution moves.

Two nearly parallel lines cross at a well-defined point that a hair's nudge sends a long way, so if the coefficients were measured or rounded the answer is simply not determined to the precision you wanted. Conditioning is a property of the problem and stability of the method, a stable algorithm on an ill-conditioned problem gives the answer the perturbed problem deserved.