To be clear, You can not have current flow without voltage and a load applied. Enter Ohms law.
If you have a 1 ohm load applied to a 5v power supply capable of delivering 10 amps or more, it will never draw more than 5amps.
If you have a 1 ohm load applied to a 3v power supply also capable of delivering 10 amps or more it will never draw more than 3amps.
If you have a 1 ohm load applied to a 5v power supply capable of delivering only 2 amps, it will not be able to deliver enough current and the voltage will get drawn down to 2v.
That said, current can be limited on a power supply but you can't arbitrary dial in x amps on a dial.
Yes, There are many linear power supplies where you can set the maximum current limit.