I once saw a particularly interesting physical manifestation of this paradox (performed I believe by Chris Bishop), where a weight was suspended from two partially elastic ropes, both attached to the ceiling. These two suspending ropes were connected in the middle by another rope. The weight was analogous to the destination, and the ropes were the roads, with the total duration of the route being analogous to the distance of the weight from the ceiling. When we removed the central road (by cutting the connecting rope), the weight paradoxically went up instead of down.

I wish I had a video of this demonstration, it really hammered in the point (for me) that this is a real phenomenon and not just mathematical trickery or electrical weirdness.