The known secret-sharing schemes for most access structures are not
efficient; even for a one-bit secret the length of the shares in the
schemes is $2^{O(n)}$, where $n$ is the number of participants in the
access structure. It is a long standing open problem to improve these
schemes or prove that they cannot be improved.  The best known lower bound
is by Csirmaz (J. Cryptology 97), who proved that there exist access
structures with $n$ participants such that the size of the share of at
least one party is $n/\log n$ times the secret size. Csirmaz's proof uses
Shannon information inequalities, which were the only information
inequalities known when Csirmaz published his result. On the negative
side, Csirmaz proved that by only using Shannon information inequalities
one cannot prove a lower bound of $\omega(n)$ on the share size. In the
last decade, a sequence of non-Shannon information inequalities were
discovered. This raises the hope that these inequalities can help in
improving the lower bounds beyond $n$. However, we show that any
information inequality with four or five variables cannot prove a lower
bound of $\omega(n)$ on the share size. In addition, we show that the same
negative result holds for all information inequalities with more than five
variables that are known to date.