Upper bound for size of minimal covers of a set Would appreciate any insight about the following regarding set covers:

Begin with a universe set $X = \{x_1,x_2,...,x_n\}$ and a set $S=\{s_1, s_2,...,s_p\}$ such that each $s_i \subseteq X$ and $\bigcup S = X$.    Consider the task of finding all inclusion-minimal covers of $X$.  (Covers are inclusion-minimal if the removal of any subset $s_i$ from the cover destroys its coverage property.)

Given the set of inclusion-minimal covers $C=\{c_1, c_2, ... c_m\}$, what is an upper bound for $\max|c_i|$, that is, the largest inclusion-minimal cover?   It seems fairly straightforward to show that one upper bound is $|X|$ (that is, an inclusion-minimal cover will contain at most the same number of subsets as there are elements in  $X$ itself).   

But can a better bound be found?

Additional information:   No restrictions are assumed for $S$.   The bound would ideally be expressed in terms of one or more properties of $S$ (in whatever form those might take).

A conjecture: If $\min |s_i| = k$, then $\max |c_i|$ (the size of the largest inclusion-minimal cover) is bounded above by $|X| - k + 1$.  This is based upon two observations:

(1) In constructing any inclusion-minimal cover, we start with any element from $S$ (call it $s_{1'}$).  As $s_{1'}$ covers a particular subset of elements from $X$, the next added cover element (call it $s_{2'}$) must cover at least one new element from $X$ not covered by $s_{1'}$.

(2) If $\min |s_i| = k$, then we can choose the first element $s_{1'}$ in our cover to have size $k$.  This leaves $|X| - k$ elements in $X$ left uncovered. By the first observation, each additional $s_i$ must cover at least one new element in $X$, so we can add at most $|X| - k$ additional elements to the cover, leading to a cover size at most $|X| - k + 1$.    

As noted in the comments, this type of bound might not be significantly smaller than $|X|$ itself.  However, I am hoping to use this in a combinatorial context, so any savings helps.  Even smaller bounds would still be useful.
The trivial bounds are that $|c| \le |X|=n$ and $|c| \le |S|=p$.

Knowing bounds on the sizes of the sets $s_i$ doesn't help much.  For instance, consider the family of sets $s_1,\dots,s_{n-k+1}$ given by $s_i=\{i,n-k+2,\dots,n-1,n\}$.  These sets are all of size $k$, and yet the largest inclusion-minimal cover has size $n-k+1$, i.e., exactly equal to the trivial bound of $|S|$ and fairly close to the trivial bound of $|X|$.  So, knowing only the sizes of the sets does not help to provide upper bounds that are much better than the trivial ones.