I suspect (obviously without proof) that while a 100-dimensional being might have a hard time directly visualizing a 101-dimensional space since all their specialized neural-equivalent-hardware might be set up for 100 dimensions that they would at least have less mathematical trouble with going up one dimension than we do. By that point the dimensions are becoming more regular in a lot of ways, whereas for us we're still stepping from something that's still often a special case (3) to something else that is also quite often a special case (4), with only the ability to visualize things that are super-special cases below us to guide us. Too much special case going on with too many properties for us to get a general understanding of dimensionality very easily.
To give one example of a special case we live in where even going to 4 doesn't help much in escaping it, we are used to the unit sphere taking up most of the volume of the enclosing cube. This turns out to be a special case for low dimensionality; in the general case it takes up a vanishing fraction of the volume. This makes sense and is perhaps even obvious if you think about it completely algebraically; if I'm picking variables from -1.0 to 1.0 for each dimension and doing SQRT(a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2...), the more I add inside that expression the less likely the resulting value is going to be less than 1.0. But even as I know this quite easily algebraically, the geometry portion of my brain is screaming "No, it cannot be! The sphere is large!", because it's too trained on special cases and generalizing those special cases into the general case incorrectly.
(Yes, the SQRT doesn't do much here since we're doing the unit circle, but it's part of the form.)