I'm modeling a bird and trying to give him a beak-opening.
I was working in smooth mode (keyboard shortcut '3') and I selected the edges of his mouth in that mode. Then, when Maya wouldn't let me separate them, I switched back to the lower display setting ('1') and realized that all of those edges weren't actually on the same edge loop - they're all over the place!
Earlier with this model, maya had also forbidden me from using symmetry, saying that my half-model was not symmetrical. I'm sure these issues are related.
The easiest solution is to use the Multi-cut tool to cut along the geometry, parallel to where you've already outlined and then deleting the geometry between the two edges. This will give you space to make the inside of the mouth with fewer maya hang-ups