Well, HutchinsonJC, yes and no. Recognize that OP already HAD created his "Ground" connection to a water pipe, and what I suggested was how best to ensure the connection at the computer end was good.
But the ambiguity here is whether or not the "water pipe" connection really is a true Ground.Many places use the water supply line where it first enters the house as a Ground. This PRESUMES that it is a metal pipe that is buried for a long distance deep in the ground so it has good electrical contact with the earth. Even if that is so, at least two other elements must be ensured. One is tha the connection of "Ground" cable to that pipe is a relly good low-resistance connection. The other is that it MUST be nade at a point on that pope BEFORE any other junctions in the piping that might be poor electrical contacts. For example, one of the more common errors is to assume that ANY water pipe in the house is a good ground, whereas there may be corroded pipe joints somewhere or even PLASTIC piping in some line that eliminates electrical contact completely! Many people trying to create a convenient Ground connection (maybe even OP here) do NOT run a Ground cable right back to the basement where the water supply line enters. Its easier to run to the nearest water pipe, but that is NOT guaranteed to be a good Ground without extensive checks.
HutchinsonJC, the hypothetical case in your second paragraph presents one possible problem, but perhaps not what you are worried about. IF the Ground line used for that "appliance" really is a good Ground, then there is no real problem between the appliance body and the water tap. It is VERY unlikely that the tap is at some voltage above Ground. The water tap itself might NOT have a solid Ground conection so that in reality it is isloated from true Ground, but unless there is some source of voltage to that water tap that's not a problem. BUT if the appliance "ground" line really is NOT a good Ground, AND the appliance is faulty so that it can supply some voltage to the appliance body, then there IS a problem pretty much the same as if no effort had been made to Ground the appliance. In such a case, even though the user falsely believes that appliance body to be safely grounded, touching both that body and a real Ground (e.g., maybe the water tap) provides a path for current flow through the human body - and likely, across the chest which is the most dangerous pathway! So in that case your fears are valid. And that is why these effots are risky - too often the people involved fail to be SURE that they really have created a reliable true GROUND.
By the way, another version of relying on a NOT good source of Ground is making a connection to the metal mounting box in the wall that holds the electrical outlet - either by connecting to the metal box, or by using the screw holding the cover plate in place. In a modern electrical system that box will be connected to a good Ground by the bare copper bonding wire in the electrical cable that runs back to the breaker panel. BUT in many older systems - especially ones that do not have a three-prong outlet installed in the orginal system - the wiring in the walls does NOT include that bare bonding wire, and the wall box is NOT Grounded - it is just floating and isolated.