An equivalent reformulation of this question: if you can cover $K_n$ by $m$ cliques, what is the smallest possible size of the largest clique? For example, if $K_n$ is covered by $4$ cliques, then at least one of them has size $\frac{3n}{5}$ (which is rather surprizing, because the edge count yields a lower bound $\frac{n}{2}$). Also this $\frac{3n}{5}$ is asymptotically sharp: construct an example for $n=5$ and make multiple copies of each vertex.

The most relevant reference, I think, is <br>
*P. Horák and N. Sauer, Covering complete graphs by cliques, Ars Combin. 33 (1992), 279–288.* <br>
They find exact estimates for some other values of $m$. If I remember correctly, these are related to finite projective spaces, as Brendan points out in his comment.

By the way, here is a funny reformulation of the result stated in the first lines: Assume each person in a country speaks some of $4$ languages, and any two persons can communicate. Then there is a language spoken by at least $\frac{3n}{5}$ of the population.