We reconsider the relationship between the bound state and the SU(3) rigid rotator approaches to strangeness in the Skyrme model. For non-exotic S=-1 baryons the bound state approach matches for small mK onto the rigid rotator approach, and the bound state mode turns into the rotator zero-mode. However, for small mK, we find no S=+1 kaon bound states or resonances in the spectrum, confirming previous work. This suggests that, at least for large N and small mK, the exotic state may be an artifact of the rigid rotator approach to the Skyrme model. An S=+1 near-threshold state comes into existence only for sufficiently large SU(3) breaking. If such a state exists, then it has the expected quantum numbers of Θ+: I=0, J=1/2 and positive parity. Other exotic states with (I=1,JP=3/2+), (I=1,JP=1/2+), (I=2,JP=5/2+) and (I=2,JP=3/2+) appear as its SU(2) rotator excitations. As a test of our methods, we also identify a D-wave S=-1 near-threshold resonance that, upon SU(2) collective coordinate quantization, reproduces the mass splittings of the observed states Λ(1520), Σ(1670) and Σ(1775) with good accuracy.