Relations between probabilities of "almost" independent random variables Let $X$ and $Y$ be two random variables, such that the (average) mutual information is very small:
$$ 0 \le I(X;Y) \le \epsilon \ll 1$$
In this case, we say that $X$ and $Y$ are almost independent. Now, can we deduce something like:
$$\forall x,y \quad \Pr[X=x,Y=y] \le \Pr[X=x]\cdot\Pr[Y=y] + \delta$$
with $\delta$ being a (possibly negative) function of $\epsilon$?

If the general case above cannot be proven, maybe we can use the following special case:
Let $X$ and $Y$ be discrete random variables, taking values from a finite set $D$ with $|D|=2^n$. Moreover, let $\epsilon$ be exponentially small in $n$, say $\epsilon = 2^{-n/2}$. 
Can we now obtain $\delta$ as a function of $\epsilon$?
 A: Let $D_{\mathrm{KL}}(P\|Q)$ denote the Kullback–Leibler divergence between discrete probability distributions $P$ and $Q$. It is well-known that the following relation holds between the KL-divergence and mutual information:
$$I(X;Y)=D_{\mathrm{KL}}(P(X,Y)\|P(X)P(Y)) \enspace,$$
where $P(Z)$ denotes the probability distribution corresponding to the random variable $Z$.
Now, consider the definition of total variation distance between discrete probability distributions $P$ and $Q$:
$$\Delta = \frac 1 2 \sum_x \left| P(x) - Q(x) \right|\enspace.$$
Pinsker's inequality gives the relation between the KL divergence and the total variation distance:
$$\Delta(P,Q) \le \sqrt{\frac{\ln 2}{2} D_{\mathrm{KL}}(P\|Q)} \enspace.$$
(The term $\ln 2$ appears since I'm measuring the entropy in bits, while the respective Wikipedia formula uses nats.)
Finally, we note that for any $x,y$, we have ($\delta$ and $\epsilon$ are defined in the question):
$$|\delta| = \Big|\Pr[X=x,Y=y]-\Pr[X=x]\cdot\Pr[Y=y] \Big| \le 2\Delta(P(X,Y),P(X)P(Y)) \le 2\sqrt{\frac{\ln2}{2}D_{\mathrm{KL}}(P(X,Y),P(X)P(Y))}=2\sqrt{\frac{\ln2}{2}I(X;Y)}=\sqrt{2\epsilon\ln2} \enspace.$$
PS: This specially appears to be consistent with an example by @whuber (see comments below the question).
