One more piece of information: I am drawing the 1D texture first, then the 2D texture.
If I reverse the order, then my 2D texture works fine, but my 1D texture does not.
And I know for a fact that my uniform flag indicating which texture I'm using is being passed down correctly, and it is using sampler1D for the 1D texture and sampler2D for the 2D texture. But, for some reason, after the first texture is drawn - either 1D or 2D - the second one in the same unit is not drawn.
Not clear why this is the case...