Aspects of the solution to the linearized water-wave problem involving a pair of surface-piercing cylinders in oblique waves and infinite water depth are examined. In particular, the solution is proved to be unique for certain geometrical arrangements and wave parameters depending on whether the wave frequency is above or below the cut-off frequency. Outside these regions of uniqueness are constructed examples of non-uniqueness using ideas developed in McIver (1996) in the normal incidence case. Although non-uniqueness examples are obtained numerically, we are able to prove the existence of non-uniqueness under the assumption that the wave obliqueness is small.