Some cars (with naturally softer suspension, such as luxury sedans) may go back up and down a little. Cars designed with stiff suspension (such as sports cars) shouldnt. SOMETIMES they will "fail firm"... meaning rather than bouncing in large motions, they will get overly stiff and will not compress well, causing a harsh ride.... as if on a bicycle or a log truck.
My old 1973 Pontiac Grand Ville is a tank and it will bounce about 2-3 times, but probably normal for a 5000 lb tank. My Mercedes will bounce a little and may need new shocks, but not bad enough to worry with now. My Lexus and Chevy are fine. They have "soft" suspension, but dont bounce with the bounce test.
If a car has excessive float and bounce, then they probably are worn and need replacing. Its been my past experience if you need new shocks or struts (whatever your car has) is to get factory OEM ones at a dealer (which will cost you MORE and downright costly on some cars), but well worth it. I have found that on past cars that cheap aftermarket shocks give too harsh and stiff of a ride. Normally you would think the cheaper ones would be the softest dampened, but they are not. They are usually some of the harshest.
Cars with air-suspension option (Lincoln Town Car, Lexus LS400, Chrysler New Yorker, Buick Park Avenue, etc) can get very costly to replace the air shocks.... sometimes $1000-$1500 each! My dads friend has a 1988 Lincoln Continental with factory air-suspension... well it did have.. they went out and it was replaced with standard struts with some modifications. My grandma had a 1988 Chrysler New Yorker Landau Mark Cross Edition, with air suspension and WOW what a comfortable car.... with pillow seats, air-suspension, you literally never knew there were ANY bumps in the road. It was like floating on a cloud... (that was the only good thing about that car... lol).
I have a Lexus LS400 without the air suspension, but it still has a soft ride, but I have rode in some with air-ride and wow, they are smooth as a cloud, but costly to replace when the fail.