Consider the outside of a straw. The surface is a curved two-dimensional space which is bounded in one direction only by the length of the straw, but is quite small in the other (the circumference of the straw). If the straw is very long and very thin then it may appear to be essentially 1-dimensional, but the surface is still two dimensional.
Nice example. Wouldn't there be easily measurable evidence of this? Imagine a flattened caterpillar like creature living on such a straw. Unwittingly, one day it might wrap itself round the straw many times. Looking back, seeing itself turned round the straw in this way, wouldn't it see slices of itself rather than a whole?
