  Fisher proved in 1940 that any $2$-$(v,k,\lambda)$ design with $v>k$ has at
least $v$ blocks. In 1975 Ray-Chaudhuri and Wilson generalised this result by
showing that every $t$-$(v,k,\lambda)$ design with $v \geq k+\lfloor t/2
\rfloor$ has at least $\binom{v}{\lfloor t/2 \rfloor}$ blocks. By combining
methods used by Bose and Wilson in proofs of these results, we obtain new lower
bounds on the size of $t$-$(v,k,\lambda)$ coverings. Our results generalise
lower bounds on the size of $2$-$(v,k,\lambda)$ coverings recently obtained by
the first author.
