The rules on this one are a wee bit complicated. If the supporting unit, let's call it A, is not supporting an attack on the unit being used to break the support, let's call it B, then the support is cut.
Just to make this clear:
France:
Army at Gascony move to Burgundy.
Army at Paris support move to Burgundy from Gascony.
Germany:
Army at Picardy move to Paris.
Army at Burgundy holds.

In this case, the unit in Picardy cuts the support provided by the unit in Paris.

However in the case Gobbledydook mentions, where the the support is for a move at the unit whose support we're trying to cut, then it gets complicated.

If we had the following moves:
France:
Army at Gascony move to Burgundy.
Army at Paris support move to Burgundy from Gascony.
Germany:
Army at Burgundy move to Paris.

Since Burgundy is trying to cut support being given against itself, it fails.

Got it so far?

Now, if a supporting unit is dislodged, doesn't matter from where, even the target of its support, it's support is cut. This means if we had the following:
France:
Army at Gascony move to Burgundy.
Army at Paris support move to Burgundy from Gascony.
Germany:
Army at Picardy support move to Paris from Burgundy.
Army at Burgundy move to Paris.
Army at Munich move to Burgundy.

Then the unit at Paris will be dislodged and it's support cut, the army at Burgundy will move into Paris, and the Munich and Gascony armies will bounce at Burgundy.

But wait! There's more!

Any unit that isn't ordered to move can receive hold support. This means a unit giving support to a move can receive hold support. Therefore if the moves were:
France:
Army at Gascony move to Burgundy.
Army at Paris support move to Burgundy from Gascony.
Army at Brest support hold at Paris.
Germany:
Army at Picardy support move to Paris from Burgundy.
Army at Burgundy move to Paris.

Then the army in Paris would not be dislodged and since the only unit attacking it is the target of its own support move, the support is not cut and the army in Gascony walks right into Burgundy.

The supports can be piled on both sides as position allows, but you get the idea, right?

So figure it this way, if the support unit is either dislodged or attacked by a unit that the support unit is not providing a support move against, its support is cut.

wonderful