We consider the free additive convolution $μ_α\boxplusμ_β$ of two probability measures $μ_α$ and $μ_β$, supported on respectively $n_α$ and $n_β$ disjoint bounded intervals on the real line, and derive a lower bound and an upper bound that is strictly smaller than $2n_αn_β$, on the number of connected components in its support. We also obtain the corresponding results for the free additive convolution semi-group $\{μ^{\boxplus t}\,:\, t\ge 1\}$. Throughout the paper, we consider classes of probability measures with power law behaviors at the endpoints of their supports with exponents ranging from $-1$ to $1$. Our main theorem generalizes a result of Bao, Erdős and Schnelli~[4] to the multi-cut setup.