A general answer cannot be given.
Here's the key:
If the motor can not run continuously, there should be a note in the manual or technical data. As for example you'll find in many small electric kitchen tools that they shall be operated for 5 minutes max. continuously, then need to cool down.
Large industrial motors are often designed for continuous operation as you can't shut down a factory regularly in short intervals to let the motors cool.
If i keep operating the motor and provide the necessary cooling does its temperature increases to a certain point and stop?
With increasing motor temperature the energy removed by thermal conductance and radiation increases. At some point the added energy from the motor's operation and the removed energy from thermal conductance and radiation will reach an equilibrium. At this point the temperature of the motor will no longer rise. As long as this point lies within the specifications of the motor all is good.
This happens with every system that dissipated energy, not only motors.