to fall between two stools

How about this phrase? is there another idiom that has the same meaning.
Thanks in advance.

"Between two stools one falls to the ground."

A donkey between two bundles of hay srarves.
A man cannot whistle and drnk at the same time.
Doing everything is doing nothing.
Grasp all, lose all.
He that grasps too much holds nothing.
He who hesitates is lost.
If you run after two hares you will catch neither.
No man can both sip and blow at once.

I had never heard the full phrase before. When I did, it was used to refer to something that fell between two disciplines or areas of responsibility. "Organising the *cleaning* of Old Street tube? That falls between two stools. London Transport claim it is the council's responsibility and Islington council claims it's just the opposite."

The original phrase seems subtlety different. It is really more of a caution than a description of a particular situation. Has any one else heard it used the way I described? I just wanted to make sure I haven't been wandering around misinterpreting for years.

Thanks,
C