The cap is designed to refuse to be over-tightened
Inside a fuel cap there are two separate jobs, and they are deliberately disconnected from each other. The thread holds the cap on the neck; the knob you turn is linked to it only through a small ratchet — a ring of angled teeth with a spring-loaded pawl riding on them. Turn the knob and the pawl drags the thread round, until the torque reaches a set value, at which point the pawl rides up over a tooth, clicks, and slips instead of pulling.
Which means the cap physically cannot be tightened past that value, no matter how strong your hands are. Every "I tightened it too much" story is therefore wrong on the mechanism, and every attempt to fix a stuck cap by pulling harder is working against a component that is not the cause. The questions to ask are different: is the clutch slipping, is the thread seized, or is something holding the cap down?
The three answers are separated by sound, and it takes five seconds. A steady click-click-click with the cap not moving at all is a worn clutch: the pawl is skipping teeth and nothing is reaching the thread, so the cap is effectively free and the knob is spinning in your hand. Complete silence and no movement whatsoever is the opposite — the whole assembly is locked, either by a corroded thread or by vacuum in the tank pulling the cap onto its seal.
Where a vacuum comes from, and why pressing down works
As fuel is drawn out, the tank would become a vacuum, so it is vented through a charcoal canister. If that vent path is blocked — a stuck valve, a crushed hose, a canister full of fuel from overfilling — the tank pulls a partial vacuum and the cap is held onto its seal by air pressure alone. That is why the cap can be immovable after a long drive and perfectly normal the next morning, and why a faint hiss when it finally releases is a clue rather than a noise.
The release for both the vacuum and a dry thread is the same, and it is not more force. Press the cap firmly down towards the neck while you turn it. That takes the load off the thread and, on many caps, disengages the pawl so the knob drives the thread directly. If it is still locked, tap the cap lightly with the heel of your hand while pressing — vibration breaks the stiction of a dry or salted thread far better than torque, and torque is what strips plastic.
When it turns but never comes out
A cap you can spin freely that never releases has already lost its thread. Plastic threads on the cap wear down against the harder neck, or a sealed cap has been forced on cross-threaded once and chewed its own thread flat. Pulling while turning usually gets it off the last few millimetres, and then you buy a cap — they are inexpensive and, unlike almost everything else on a car, they are a genuine consumable.
Before fitting the new one, look inside the neck with a torch. Debris in the thread — grit, a flake of rust, a piece of the old cap's seal — will chew the new cap exactly as it chewed the old one, and that is a five-minute fix with a rag and a small brush. If the neck itself is deformed or the seal seat is corroded through, that is not a cap problem and it needs a workshop, because a filler neck that will not seal is an inspection failure and a fuel smell in the cabin.