On the other hand, can you say definitively that we possess no dimension higher than three or four ourselves? I saw a discussn of Flatland somewhere here earlier. In that book, it was said that A. Square did in fact have a third dimension to him; it was just extremely thin, so that it appeared nonexistent. Could the same be true for ourselves relative to other dimensions?

Again, you can't say this is true. You can't say it's not true. But, it does render your use of 'impossible" earlier more or less incorrect.

(By extension, and using the same book's logic, we can exist in lesser dimensions. The only issue is, only a cross-section of ourselves would exist on that plane/line/whatever, while the rest of us is outside of it.)