Not necessarily. I had the exact same problem, mouse would only go left and right, but not up and down. Blown fuse. Seems like that fuse is on one of the axis lines. Why? I haven't the foggiest.
Actually, the fuse is in the 5v power line to the mouse port. If it blows, there's no power in the mouse and neither axis works. Joysticks and mouse buttons work, cuz they're just switch closures. The fuse is only in newer motherboards... older ones could burn out traces on the board. I did see one where the mouse was sensitive to how tight the mounting screws next to the mouse port were... I've also seen a poorly seated Agnus chip do this.
If you've got a voltmeter, you can check for 5v at the joystick ports... it's been a while, but i think it's between the middle two pins on the bottom row of the connector. I used to check a lot of these, so I made a tester by wiring an LED and 220 ohm resistor to an old DB9 connector.
