Anytime you sit for an extended period, you could subject your back to pressure. Driving for hours can distract you from the pain your back is going through, which is all the more reason to prepare for it before it happens.
Unlike plane travel, though, you have control over your environment when you're in a car, and you can make adjustments during the trip to stay comfortable. That doesn't just include the interior, but the way you drive as well. Serious bumps in the road can aggravate medical conditions and make driving more painful than it needs to be.
Obviously, there's only so much you can do if the road you're driving on is in bad shape. That's why planning ahead is crucial. You may want to inspect the vehicle you'll be traveling in beforehand to make sure it's ready for a long trip. Specifically, pay attention to the tires and shocks before you decide to take a particular car out for a spin.
That's fine if you know what to expect, but what if you don't? Pain over speed bumps might not actually be from your back, since multiple sources trace this to appendicitis. If you've never felt this before, this could be the wake-up call you need to go see a doctor. Distinguishing between symptoms could then give you a better idea of how to take future trips without more pain.
As always, you'll need a medical professional to help you figure out the source of back pain for sure. Schedule time with one soon after feeling back pain on the road and start reducing the chance of pain in the future.