Arduino Nano joystick x2 / potentiometer
Yes, this should work. The potentiometer slider will source about 5mA and each joystick axis about 50mA. In total this gives you 205mA. Here it is important, that your power source can provide enough current. If you are using an USB port from your computer, this would result in 500mA maximum that can be drawn from the bus. Since this is shared with the Arduino you can externally use about 400mA

For more information about current ratings refer to this question

EDIT: Since your previous resistance values were not correct, I edit my answer.

When all potentiometers have 10kOhm resistance (and this seems to be more likely the right values), each one draws about 5mA from the 5V pin (You can calculate this with Ohm's Law U = R * I, which gives you I = U / R. So you have to divide the voltage by the resistance.). So you aren't getting anywhere near the problematic currents. These potentiometers are designed this way. They don't want to draw a high amount of power, but only setting a certain voltage to the output pin.