Eavesdropping.

http://en.wikipedia.org/wiki/Operation_Ivy_Bells is An example. It is rumored they can do that on fiber, too (http://cryptome.org/nsa-fibertap.htm)

But that doesn't explain why they were cutting it.

I think the many-parent's idea is: cut it in one place in an obvious way, then while the cable is known to be cut, insert your splice in another place. That way no-one notices the few hours' disconnection while you put the splice in.