Let $X=x_1\ldots x_n$ be a random variable. Assume that every $x_i$ takes values in $\{0,1\}$.
Assume also that for every $I \subseteq \{1,\ldots, n\}$ the Shannon entropy of random value $X_I$ [if $I = \{i_1, \ldots, i_k\}$ then $X_I = x_{i_1}\ldots x_{i_k}$] is equal to $|I| \cdot h(p) + O(\log n)$ for some $p$, where $h(p)= -p\log p - (1-p)\cdot \log (1-p)$.
What can we say about $X$? If the term $O(\log n)$ is just zero then all is simple: the variables $x_1, \ldots, x_n$ are independent and have the same entropy. So, $X$ has a Binomial distribution with parameters $n$ and $p$ that is shifted by some vector in $\{ 0,1\}^n$.
In general case I think that $X$ can be approximated by a mixture of several random variables with Bernoulli distribution with the same distribution. More precisely my conjecture is the following: there is a random variable $Y$ that is a mixture of poly($n$) Binomial distribution (with parameters $n$ and $p$; that are shifted by some vectors in $\{0,1\}^n$) and polynomial $q$ such that the following is true for 99% (according to probability measure $X$) $z \in \{0,1\}^n$: $\Pr[Y=z] \cdot p(n) \ge \Pr[X=z]$.
