A friend of mine is pretty sure Kenneth Copeland is a part of a fourth dimensional angler fish. He's just out here looking vaguely human and teaching the Bible just wrong enough so that instead of Jesus coming back it's going to be some nightmarish horror.

Also, he eats a pet every few days. Not because he needs meat, but because he feeds off the suffering of children.

We actually kinda do perceive a fourth dimension: time. Sure, we infer it from our memories and come up with cause and effect relationships to help us understand it. But we do know it's there.

If there are higher dimensions, say the extra seven asserted by String Theory, then we have breadth (thickness?) along each axis that is non zero. The higher-order string theory dimensions (which communicate particle information like gravity) are tightly rolled up.

Brian Greene uses the metaphore of an ant on a wire who can move along the wire freely, but can't go far laterally. They may be so small that our quantum bits can't drift anywhere, so our liver doesn't abandon us drift along a high-level axis.

If there are flat higher level dimensions, then either a force or some kind of membrane would have to exist to keep our blood from leaking.

That said, when we have pure elements, or even pure minerals or chemicals, they retain the same density (mass to volume, sometimes affected by temperature) which suggests nothing is hiding away in other dimensions whenever we take measurements. If there is room along higher axes for unseen activity, it doesnt bug us enough to work out consistent properties.

We don't! When I was younger I had a theory that the brain is a 3D representation of an organ that exists in a higher dimension. Granted, I had (and still have) no relevant expertise to properly speculate on how that could work, but it was fun to think about.

Because we see no evidence of a 4th spatial dimension. So if there is a fourth dimension, our universe doesn't seem to have access to it.