Permit me to repeat this example from a related MO question, based on a paper by
George Tokarsky, "Polygonal Rooms Not Illuminable from Every Point"
[*Amer. Math. Monthly*, 102:867-879 (1995)]. Here is a figure from the MathWorld article
on the topic:

These are rooms all of whose walls are mirrors but which have a pair of points $x$ and $y$
such that if a light is placed at $x$, point $y$ is dark.
It is an unresolved conjecture that, at least in rational polygons (angles rational multiples
of $\pi$), the number of dark points for any light position is finite.
If that conjecture were true, then it seems plausible that it would be possible to place additional internal mirrors to
cover those points and still illuminate the remainder of the room.

**Update** (

*13 May 2015*).
The "unresolved conjecture" I mention above has been settled.
I found the reference in
Alex Wright's 2015 paper, "From rational billiards to dynamics on moduli spaces."
(

arXiv abstract):

Lelièvre, Monteil and Weiss have shown that if $P$ is a rational polygon,
for every $x$ there are at most finitely many $y$ not illuminated by $x$.

The reference is:
Samuel Lelievre, Thierry Monteil, Barak Weiss.
"Everything is illuminated." 2014.
(arXiv abstract.)