  Let S $\subseteq$ N be a numerical semigroup with multiplicity m, conductor c
and minimal generating set P. Let L = S $\cap$ [0, c -- 1] and W(S) = |P||L| --
c. In 1978, Herbert Wilf asked whether W(S) $\ge$ 0 always holds, a question
known as Wilf's conjecture and open since then. A related number W0(S),
satisfying W0(S) $\le$ W(S), has recently been introduced. We say that S is a
near-miss in Wilf's conjecture if W0(S) < 0. Near-misses are very rare. Here we
construct infinite families of them, with c = 4m and W0(S) arbitrarily small,
and we show that the members of these families still satisfy Wilf's conjecture.
