Power supply conundrum
It sounds like perhaps your voltage is set to below 1.0V. By Ohm's law (V=IR), you only need 1 volt potential difference to create a current of 1 amp across a 1 ohm load.  What is the voltage across the output terminals of the power supply?

As an aside, it is probably more appropriate to say "The power supply can supply up to 3A." Power supplies often have two modes: voltage source or current source. In the voltage source mode (more common, usually) you set the voltage you want and draw the current your circuit draws up to the limit that the power supply can supply you (3A in your case). The second (current source) mode is probably more appropriate for the "pushing" analogy, since in this mode, the current source sets the voltage across its output terminals to create the desired current, again up to a limit.