Laboratory core-sample measurements do not record elastic constants directly; they record traveltimes from which the elastic parameters must be deduced. But do the measured traveltimes represent group velocities, phase velocities, or something else? For propagation down symmetry directions group and phase velocity are the same so there is no ambiguity. However, propagation in nonsymmetry directions is key to determining a complete set of elastic constants. In nonsymmetry directions there is no guarantee the energy radiated from the source will travel straight up the core to the receiver; the leading planar portion of the wavefront may crab sideways and partially or completely miss its desired target. Our model results show that unless the miss is complete picking first breaks will still give a good phase-velocity arrival time, the amount of additional error due to the miss being less than the normal random measurement errors of one or two percent. Unfortunately it is still true that even a small error of only one or two percent in a crucial nonsymmetry-direction phase-velocity measurement can result in a significant error in the derived value of C13, especially for qP waves. If good qP and qSV nonsymmetry-direction measurements are both available, however, the error bounds on C13 can be considerably improved.