My professor said (hesitantly)
$\textit{If it works in 1-D, it is likely that it also works in N-D}$
Hesitantly because, she remarked, it is not true for everything (which is expected).
After some research, I found one such example:
In 1-D and 2-D, if you start at the origin and continually do random unit steps in each of the cardinal directions (north, south, east, west, etc.), you will return to the origin with probability one. This fails for dimensions three and higher, in fact, you return to origin with probability zero in those cases.
What are some other striking examples of this?