How can you add a Uniform random number to every point
in an uncountable bounded set, and end up with only
finitely many answers?

Let U be Uniform(0,1), and for any x on the line, let
Y(x) = min(i+U: i+U>x, i in Z) (Z being the integers).
It's easy to see that Y(x)-x is Uniform(0,1), and that
for any bounded set A, {Y(x): x in A} is a finite set.

Pictorially, drop a unit-spaced grid down on the line,
with a random location. Map each point x in A to the 
nearest grid point Y(x) to the right.

This is called the Uniform Shift Coupler, and is an ingredient in
practical methods of coupling from the past. See David Wilson's
bibliography.