Paddle controllers on the Atari 2600 use a variable resistance in a resistor-capacitor timing circuit and then measure how long that circuit takes to switch. The time will vary from a few scan lines to a couple hundred scan lines, depending upon paddle position, and software will typically need to scan the controller position during the video frame. For games that use one paddle, its position will typically be sensed "from scratch" on every frame; games that use more will typically read paddle positions "in rotation". Games can typically sense any amount of paddle movement per frame, and many will respond directly to paddle movement.
In my own partially-completed variation of Arkanoid, I used a different approach to reading paddles which limited their speed. Rather than resetting the timing circuit at the start of each frame and checking on every other scan line whether it had timed out yet, I selectively set or clear the "reset" signal to the timing circuit on every 8th scan line, based upon the previous paddle position, and then I check for a timeout on each line during (IIRC) a 48-line area near the bottom (below the bricks). This reduces the overhead required to read the paddle during the displayed part of the frame (freeing up CPU time to display more content) but limits paddle motion to 32 pixels/frame. Even though the speed is not unlimited, it's still pretty fast (if memory serves, my version of the game was 144 pixel across, so about 1/10 second would let the paddle fully cross the screen).
I don't know if any other paddle games used this approach, but it reduces the CPU overhead required for paddle sensing at the expense of maximum paddle speed.