I would be very surprised if you receive a "conceptual" answer to this problem- though I would be delighted to be proved wrong. Regarding your last comment, there have been examples recently where computational evidence has indicated that human intuition about the size of cohomology groups was probably faulty, being based on limited evidence.
Regarding the comment about the bad general bound for the size of the Schur multiplier of a finite group, it can get quite big for $p$-groups, as you no doubt know. If my memory is correct, an elementary Abelian $p$-group of order $p^{n}$ has Schur multiplier of order $p^{n(n-1)/2}$, as is well-known.