There's clearly no possibility we'll experience anything else
("having" more or less dimensions suddenly, or experiencing a point with no surrounding open set")
In contrast to this established view
M-theory suggests that we may be
constrained to a four-manifold embedded
in a higher dimensional space
(constrained on the Brane).
(for an early proposal see hep-th/0001113
by Akama et al.)
In such models gravity is free to propagate
into the higher dimensional space.
This has observational consequences. The
Einstein field equations have an extra term.
It should be observable in two ways.
Directly, gravitation on length scales
of a few millimeters should deviate from
Newtonian gravity. It has implications
for the evolution of Inflation in the early Universe.
Should such effects be observed we would discover that
our world is stranger than heretofore thought.
This month's (August 2000) Scientific American
carries an article about it, but I have not read it
and cannot vouch for it.