Ogasawara School of Etiquette considers that "It is best not to step on the borders but it is still permitted to step on them." Because there will be a possibility to step on the borders in a room with dozens of tatami mats even though one might adjust his/her step. Well then, how did this "not allowed to step on borders" etiquette come about? This etiquette were established at the times of Ashikaga feudal government, when Ogasawara School had almost settled its ideas.
In those days, they carried the meals or plates for Shogun, masters, or guests on a stand. They carried it raising at the height of their eyes so they won't breath on the food. As you have already realized, in this way, they are not able to see their steps. They considered best to walk one tatami horizontally with three steps and a half (men) or, four or five steps(women).
Also, there is another theory. Since they distinguished tatami for the noble men between tatami for the commoners, they believed that it is a disturbance of the sacred border when they were stepped on.
Japanese ancestors introduced this order of space which equals to mental privacy, which they tended to protect. Therefore, by combining both respect for each individual, and the rational idea of preventing tripping over the tatami border, create an etiquette of not stepping on the borders. We watch our steps when walking and it is best not to step on borders but if we are too careful, one might stagger and his/her move could turn out very stiff. Staggering foot shows the disturbed mind meaning the etiquette is not closely linked with one's soul. And this action does not practice the true heart of the Ogasawara. It is important to understand the "soul" of courtesy and not just the form of it.