Is there any legitimate reason you would NOT connect the ground wire on a circuit?
No, there is no reason not to have the ground connected. Even with armored (BX) cable, you should connect the ground. There is an older style of armored cable that uses a wide metallic strip (inside) as the ground which is a bit flaky since it's hard to get a good connection (it is just supposed to connect with the box connector). The actual armor though is not designed for grounding, and should not be used or relied on as a ground (though in most cases, the electrical connection exists for it to work as ground).

Let me be clear though -- you should check what is on the other side of the wire, and make sure it actually is grounded (or, possibly, not connected -- which you can fix), and that it is not being used for some other purpose. Just a red flag for me: when someone snips off the ground it means they didn't know what they were doing, so don't assume that anything is correct.

One thing to look for - there should only be one common ground in your house, and everything should be tied together. If you have, for example, two separate ground rods and circuits mixed, you can have ground loops and noise issues, especially if you have sensitive electronics (TVs, amplifiers, computers) connected. All grounds should be connected together at or near the main panel.

On the same topic, you should have a ground wire connecting your electrical grounding wire to all copper pipes in your house, and to any gas lines. Be careful with placement, as often you need a jumper wire to go around a non-conductive water meter to connect both sets of pipes together.