Plaster, Bryan was trying to let you know that you were using the words "ungrounded system" incorrectly. That is a carefully defined phrase, and it does not mean the thing you are describing. If the house you are talking about has a ground rod, and I think it had better or you have an entirely different problem, then it is a grounded system.
You are asking about houses (like mine) that did not originally have a separate (bare copper or green insulated) ground wire. The correct term for that wire is the "Equipment Grounding Conductor," or ?EGC.?
To address your first statement, you can add a GFCI receptacle to a system that does not have a ground wire, and the GFCI protection will work. You can connect more receptacles downstream, and they will also have GFCI protection. But that is not the same as having, as you called it, "sufficient grounding protection." What it has is the ability to detect the flow of current through a human body, and it will trip if it sees that current. But if there is a wiring failure inside a piece of equipment, such that the outer metal case of that equipment becomes energized, and if no person has yet touched the equipment, the fact that there is no EGC will mean that the breaker will not trip. So the difference is that in a grounded system, (1) The breaker will trip before a person has a chance to touch the equipment, and (2) If a person was touching the equipment when the failure occurred, the breaker would trip before the person could be harmed. On the other hand, in an ungrounded system with a GFCI receptacle installed, (1) The breaker will not trip when the internal failure occurs, and (2) If a person is touching the equipment when the failure occurs, the GFCI will trip to protect the person, but the breaker will not trip.
To address your second statement, if you install a GFCI receptacle in a system that does not have the ground wires (the EGCs), a GFCI tester will (incorrectly) report to you that the GFCI receptacle is not working. That is because, as you pointed out, the tester inserts a small fault, so that current will flow through the EGC, which would be sensed by the GFCI receptacle, and which would cause the GFCI to trip. But if there is no ground wire, then the tester will not be able to insert that small fault, and no current will flow through the tester. As a result, the GFCI will not trip, and you may wind up (falsely) concluding that the GFCI is not working.
