Lab power supplies generally have a constant current limit, so if a motor tries to draw more the supply will simply lower its output voltage to keep current down to the limit. This is different from switch-mode supplies used in computers etc., which usually cut off completely or 'fold back' to a very low current upon detecting a 'short circuit'.
How much 'inrush' current a motor draws depends firstly on the type of motor, and secondly on the particular construction.
A brushed DC motor's stationary current draw is limited only by its internal resistance, which is usually very low. Stall current is typically 3-5 times higher than the current draw at rated output, but can be as much as 10 times higher for very efficient motors. In a permanent magnet or shunt-wound DC motor, torque is proportional to current. If the power supply's output current is sufficient to get the motor spinning it will still work, but could be slow starting up. The associated momentary voltage dip could also affect other circuitry being powered by the supply.
Brushless DC motors are always operated through a controller, which usually limits startup current and/or acceleration to avoid damaging itself when starting the motor. Some controllers also have low voltage detection. This may cut the motor completely, or just reduce power output until the input voltage comes back to normal.
A stepper motor draws about the same current (or less) when holding as when stepping, so 'stall' current isn't a problem. Stepper motors also have very high inductance, so the 'inrush' current rises relatively slowly.