One aspect of relativity that used to bug me was a thought experiment that I came up with similar to the one ariels describes above. I node it here mainly so that I don't feel like I've completely wasted my time in thinking about it for so long:

Imagine a lighthouse situated at a given distance from a wall of infinite length. At t = 0, the beam of light is perpendicular to the wall.

  L                                               angle = 0

The beam then begins to rotate, sending the point of light made by the beam along the wall.
  |   /      ~         _  -
  |  /    ~      _  -
  | /  ~   _  - 
  |/~_  -
  L                                    angle = 30, 45, 60...

As the above diagram shows, the point where the beam of light hits the wall begins to travel faster and faster, even though the rate at which the beam rotates is constant. The reason for this is that the distance travelled by the point of light is calculated using the tangent function, which approaches infiniti as an angle approaches 90 degrees. Eventually, as the beam of light becomes almost parallel to the wall, the speed of the little point of light where the beam hits the wall should also approach infiniti, and therefore appear to be moving faster than light along the wall, right? Well, not really. In fact, the photons that make up the beam of light are only travelling at the speed of light themselves, which affects when they actually hit the wall. The photons leaving the lighthouse are therefore spread out in an ever-expanding spiral:
              *                            *
                 *                           *
*                  *                          *
--*---------  ------*------------  -----------*------------------
   *                *                         *
L*            L    *               L         *
               * *                 *       *
                                      * *

So how do we figure out how fast the point of light is moving? Well, like all good scientists, we substitute a simpler model because we're lazy. Imagine instead of a rotating beam of light, we release a single burst of light in all directions at t = 0. In this example, the point of light will certainly travel no slower than in the previous example, as all of the photons will be released at the same time, instead of having to wait until the lighthouse has rotated. The model should now look like this (dubious ASCII skills notwithstanding):
            *                                      *
                 *                                   *
 *                  *                                 *
    *                *                                 *
------*---  ----------*--------  -----------------------*---
       *               *                                *
 L     *    L          *         L                      *
       *               *                                *
      *               *                                 *
    *                *                                 *
 *                 *                                  *
As the third figure shows, eventually the point of light on the wall will be travelling at about the same speed as the burst that created it - namely, at the speed of light - but no faster.

That said, it should be possible to create a point of light that appears to move faster than the speed of light - if the lighthouse beam began parallel to the wall and rotated so that the beam moved in the opposite direction from that of the first example, the point of light created by the beam should appear to move faster than light (and indeed, at a theoretically unbounded speed). Of course, to put this into practice you would need an infinitely long wall1, but given that it's simply a matter of getting photons to strike specific points at specific times, it wouldn't be that hard to kluge something similar.

1ariels points out that the infinitely long wall is only necessary if we want the point of light to move infinitely fast - in order to go faster than light, a really, really, REALLY long wall would suffice.