I don't know about full rank but a nice lower bound on the rank is given in the following paper:

N. Alon, H. T. Hall, C. Knauer, R. Pinchasi and R. Yuster, On graphs and algebraic graphs that do not contain cycles of length 4, J. Graph Theory 68 (2011), 91-102.

There is a preprint version.

Let $G$ be a bipartite graph with $m$ and $n$ vertices in its partitions. Let $E$ be the number of edges and let $s(G)$ be the number of $4$-cycles in $G$.

Also, denote by $\rho(G)$ the following regularity measure:
$$
\rho(G)=\frac{4E^2}{n\sum{d_i^2}},
$$
where the $d_{i}$s are the vertex degrees.

Also, for a symmetric real matrix $M$ denote by $\rho(M)$ the following measure of the regularity of $M$'s spectrum:
$$
\rho(M)=\frac{trace(A)^2}{n \cdot trace(A^2)}.
$$

Then Theorem 6 in the above paper states that:
$$
rank(A) \geq \rho(AA^T)m,
$$
and
$$
\rho(AA^T)=\frac{E^2}{4E^2/(\rho(G)(m+n))-E+4s(G)}.
$$