Suppose $\{v_1,v_2,v_3,\ldots\}$ and $\{w_1,w_2,w_3,\ldots\}$ be two countably infinite Hamel bases of the same vector space. Must there be infinitely many values of $n$ for which $\operatorname{span}\{v_1,\ldots,v_n\}$ $= \operatorname{span}\{w_1,\ldots,w_n\}$?
Later note: Shortly after posting this I wondered why I would think such a thing, and thought of deleting the question. Then it occurred to me that maybe this is true with a simple additional hypothesis, but I'm not sure what it is. Maybe more later?