I'm not clear on what you are asking ("doesn't drop below 1A"?). But there are basically two short circuit protection schemes used with power supplies, based on whether they are switching supplies or linears.
Switching power supplies monitor current (either on a cycle-by-cycle basis, or longer term), and modify their switching behavior to pass much less energy through when there is a very heavy load on the output. They can do this, because the switching element does not dissipate much current when it is either off or on.
Linear power supplies generally use "fold back current limiting", where the output current drops to a low value, until the heavy load is removed from the output. This is done because the linear pass element of the linear power supply generally cannot support a heavy overload output current and a low output voltage, because all of the input-output voltage drop is across the pass element. So even running at it's max rated output current is generally not possible when the output voltage is pulled low by a heavy load. The increase in power dissipation by the pass element is generally too much like that, so the foldback behavior to low output current in this condition saves the pass element.
You can google these terms for more info (or use wikipedia.org), or ask follow-on questions in this thead.