We study the problem of determining upper bounds on the maximal binary rank, $b_d$, of a $0,1$ matrix with real rank $d$, for small constant $d$. We prove a general upper bound of $b_d \leq \left\lfloor \sqrt{d2^d + 2d}\right\rfloor -1$. We also develop a general dimension reduction technique of bounding $b_d$ using known upper bounds on $b_{d-1}$ and $b_{d-2}$, as well as prove interesting general structural results on extremal matrices with binary rank $b_d$. This allows us to get improved non-trivial upper bounds on $b_d$ for constant values of $d$, which imply tight bounds on $b_d$ for $3 \leq d \leq 5$. Our results also imply bounds on the equivalent problem of finding the minimum number of bicliques needed to partition the edges of a bipartite graph whose reduced adjacency matrix has real rank at most $d$.

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