Actually, this is quite possible, even probable.
Imagine one is in Rome, circa 100 B.C., and hires onto a boat carrying something to, say, present-day Portugal.
Once past the Straits of Gibraltar, things get messed up; perhaps a storm, perhaps an incompetent navigator, and one is way out in the deep deep blue Atlantic. Many on board die of injury or starvation or murder, but some still cling onto life.
Then some months later, one beaches on Texas.
One is Roman, remember, and all this is new.
One manages to survive somehow, and to leave traces of his existence.
Such things probably happened thousands of times, probably 99%+ of them evaporating in the mists of history (the boat sunk, or all aboard died of thirst and starvation, usually), but enough times that surely in two or three or four or more instances, someone survived, but lived out his life (whether a few days or a few weeks or a few months or a few years) having absolutely no idea where he was, and how to get back home.