A balls into bins problem with combinatorial constraints We are given $m$ balls and $n$ bins, with $m \ge n$. Each bin can contain at most $c$ balls (we assume that $c$ is an even integer). In a sequential fashion, at each time step, one ball is placed into one bin selected independently and uniformly at random from the set of non-full bins remained (i.e. the ones containing less than $c$ balls). 

Question: Given a positive integer $n'\le n$, what is the minimum number of balls that it is necessary to place into bins in such a way that the expected number of bins containing at least $\frac{c}{2}$ balls is equal to at least $n'$?
(What about the minimum number of balls for having that the expected number of full bins is equal to at least $n'$?)
 A: (Not a full answer, but too long for a comment.)
This is more or less the generalized coupon collector problem.  The only thing is that in your model, you stop collecting coupons of types that you already have $c$ of.  You could probably convert it exactly into the coupon collector problem by doing the following...
Coupling model: First drop balls into bins uniformly at random, and stop after a while (this is classical coupon collector).  When you stop, then remove balls from all the bins that have too many until no bins are over capacity.  This has the same distribution as your model, and it sounds easier to analyze.
In the classical model, the number of bins with enough balls (if there's no issue about capacity) is very well understood.  And the number of "excess balls" that you'd have to throw away to go from the classical coupon-collector model to your setting is hopefully something you can also control sufficiently well.
So hopefully that coupling will give you whatever you might want.
