If you have bypassed the GFCI receptacle and still have problems, then there must be a loose connection between the GFCI and the breaker. Here are the places to look:
At the GFCI, are the wires connected by a screw, or by push-in? The push-in connections are unreliable - I have had a receptacle "fry" because of poor connections, and it was installed by an electrician. Installers like them because they are fast, but I always go with the screw connection.
At the breaker, is the breaker making good contact with the power bus? If the bus is corroded, even a new breaker might not make good contact. Try plugging the breaker into a different slot, if there is one to spare.
Also in the panel, is the neutral (white) wire screwed down tightly?
Are there any other outlets between the panel and the GFCI receptacle? If so, make sure the twist-on connectors are tight, and that the splice is not made through the receptacle.
Note that loose connections will not trip a GFCI. Neither will short-circuits between hot and neutral - they will trip the breaker in the panel. Also, nothing "upstream" will trip a GFCI. And once a GFCI is tripped, it won't reset itself.
As well it should.
Normally, loads are connected between hot and neutral. Appliances are not supposed to connect hot or neutral to ground; ground is only a shield.
The GFCI compares current on the hot and neutral wires. They should be the same. If they are not, current has found another route, possibly through the grounding system (which isn't supposed to happen) and potentially through some poor human.
Circuit testers are trying to test whether ground is connected... cheaply. They mis-use "hot" as a power source, by connecting a light bulb between hot and ground. If ground is connected correctly, this will light.
In other words, it intentionally creates a hot-ground fault (by sticking a light bulb there). This is exactly the condition GFCIs are designed to detect.
I'm not talking about any ground-fault-test the tester may also have.
So why do testers often work? With a perfect GFCI, they wouldn't work. I suspect it is because GFCI's have detection thresholds above zero, and that is often enough for these testers to "get away with it". I even think there may be a tacit agreement among manufacturers for this, but obviously, lower sensitivity impinges safety. Remember a shock which only stuns you can kill you with secondary effects like falling or drowning.
So either your GFCI is pretty good, or your tester is pretty bad.
Best Answer
Yes, anything connected to the LOAD terminals of a GFCI device is GFCI protected. So a ground-fault anywhere on the circuit "after" the GFCI device, would cause the device to trip. This will cause the receptacles on the GFCI device, and all the devices attached to the LOAD terminal to lose power.
Connecting GFCI devices on the LOAD side of a GFCI device, should cause no problems other than wasting money.
If you wired three GFCI devices in line as you suggested, and then pressed the test button on the last GFCI in line. The previous GFCI devices should not trip.
Internally, a GFCI looks similar to this.
So when it's set and the test button is pressed, it looks like this.
When the test button is pressed, current flows through the test button, through a resistor, around the current transformer (CT), and back to the grounded (neutral) terminal. I've highlighted the current path in yellow.
So as you can see, you'll have 6 mA on the ungrounded (hot) terminal, and also 6 mA on the grounded (neutral) terminal. This GFCI device tripped, because the return path of the test circuit bypassed the CT. The current will not bypass the CT in the other GFCI devices, so they should not detect the test ground-fault.
This blog post might help you understand how GFCI devices work.