 Rank bounds for a sum of linear maps

The image of $A+B$ lies in $\mathrm{im}A+\mathrm{im}B$. The dimension formula for a sum of subspaces gives the upper bound. Applying it to $A=(A+B)-B$ and then interchanging $A,B$ gives the lower bound. Both extremes are sharp: cancelling diagonal blocks gives the difference of ranks; diagonal maps with supports covering the whole space and positive coefficients give full rank when the sum of prescribed ranks is at least the dimension.