An $ϵ$-almost covering system is a set of congruences $a_i(\mathrm{mod}{n}_i)$ for distinct moduli $n_1<\dots <n_k$ such that the density of those integers which satisfy none of them is $\le ϵ$. Is there a constant $C>1$ such that for every $ϵ>0$ and $N\ge 1$ there is an $ϵ$-almost covering system with $N\le n_1<\cdots <n_k\le CN$?