Relation between Hamming distances of columns and rows

You're given a $0-1$ $n\times n$ matrix such that for every distinct columns $C_i$ and $C_j$, $d_H(C_i,C_j)\gt 2t$ for some $t$. What could be said about the Hamming distances of the rows? It it true for instance that there are two distinct rows $R_k$ and $R_l$ that satisfy $d_H(R_k,R_l)\lt n-t$?

The condition on the columns implies that $2t < n$. Suppose that the condition on the rows is not satisfied. The rows therefore form a code of minimal distance at least $n-t > n/2$ of size $n$. The Plotkin bound implies that unless $n-t$ is extremely close to $n/2$, such a code cannot exist. If $t$ is very close to $n/2$, however, the columns form a code of minimal distance close to $n$, which is again ruled out by the Plotkin bound (except perhaps for very small $n$).
I'll let you fill up the details and check for which $n$ this argument works.