  Let $(K_{n})_{n=1}^{\infty}$ be the optimal constants satisfying the
multilinear (real or complex) Bohnenblust--Hille inequality. The exact values
of the constants $K_{n}$ are still waiting to be discovered since eighty years
ago; recently, it was proved that $(K_{n})_{n=1}^{\infty}$ has a subexponential
growth. In this note we go a step further and address the following question:
Is it true that \[ \lim_{n\rightarrow\infty}(K_{n}-K_{n-1}) =0? \] Our main
result is a Dichotomy Theorem for the constants satisfying the
Bohnenblust--Hille inequality; in particular we show that the answer to the
above problem is essentially positive in a sense that will be clear along the
note. Another consequence of the dichotomy proved in this note is that
$(K_{n})_{n=1}^{\infty}$ has a kind of subpolynomial growth: if $p(n)$ is any
non-constant polynomial, then $K_{n}$ \textit{is} \textit{not} asymptotically
equal to $p(n).$ Moreover, if \[ q>\log_{2}(\frac{e^{1-1/2\gamma}}{\sqrt{2}})
\approx0.526, \] then \[ K_{n}\nsim n^{q}. \]
