  Seshadri constants, introduced by Demailly, measure the local positivity of a
nef divisor at a point. In this paper, we compute the Seshadri constants of the
anticanonical divisors of Fano manifolds with coindex at most $3$ at a very
general point. As a consequence, if $X$ is a nonsingular Fano threefold which
is very general in its deformation family, then $\varepsilon(X,-K_X;x)\leq 1$
for all points $x\in X$ if and only if $\vert-K_X\vert$ is not base point free.
