First, I should've put “constant and full attention” there. My phrasing was off. As to why I came to this conclusion:
1. People often ride for a few hours without breaks. My assumption is that it's not possible to pay constant and full attention throughout this time—our minds are bound to wander.
2. The pattern I usually observe is when the driver, depending on road situation, varies the degree of attention they pay to the driving. At one time they're driving relaxedly and multitasking (talking to a passenger or someone on the phone—by definition they're not paying their full attention at this time), while at another they're more or less concentrated on the road.
Disclaimer—I don't myself drive a car, but I often observe people driving. I've observed both experienced drivers who drive frequently and more beginning drivers, the pattern described under (2) is the same.
Update: the comments here, coupled with my observations, make me believe—roughly—that a) drivers don't (can't) pay their full and constant attention to the road, and b) many are in denial of (a).