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pk - re sorting out the problem you describe / demonstrate with the cube (marcin's correct re what to do to avoid it in the first place - this might help you sort it out if / when too late)

Assuming you've done the stuff with the cube , so you've got 2 'hollows' with a thin wall between them.

this 'wall' is, in fact, 2 faces - both occupying the same space ... and sharing the same 2 verts at the top -but not (joined at) the bottom.
To see this more clearly, select one of the edges at the bottom of the wall and move it away (a little) from the wall - ie towards the outside of the cube.

The 'wall' will then have one vertical face and one at an angle (the one you moved) ... showing they're not joined at the bottom.
Select both wall faces and dissolve ... you'll end up with a face with 6 edges / verts
press v to change to vert mode / select the 6 verts

De-select the top 2 verts - just leaving the 4 at the bottom of the 'hollow'
Connect these - c and you now have proper geometry back ... dissolve unwanted edges as necessary.

For this particular cube based example - even if you used extrude region > y for the 2 faces, you'd end up with problems of a similar nature at the sides.

These can be solved in a similar way if you approch things logically.

If you find yourself in a situation where elements / verts appear to be on top of each other - don't be afraid to move stuff around in order to be able to more clearly see exactly what the nature of the problem is - it's also generally pretty obvious which way stuff should be moved to clarify matters.

You can use face | dissolve to form a really weird face with many verts ... that you can then re-form to the shape you want, by connecting verts - try it on a sphere, say.

pp