(related question : most general way to generate pairwise independent random variables?)

Let $X_1,X_2,X_3,X_4$ be four random variables with standard Bernoulli distribution (i.e. $P(X_i=0)=P(X_i=1)=\frac{1}{2}$ for $1 \leq i \leq 4$) and such that any two of those four variables are independent.

Then the fourtuple $\overrightarrow{X}=(X_1,X_2,X_3,X_4)$ takes at least one of the three values $(0,0,0,0),(0,0,0,1)$ or $(0,0,1,0)$ with positive probability.

This surprising fact can be shown by brute force: what we have is essentially a system of linear equalities and inequalities in 16 variables corresponding to the distribution of $\overrightarrow{X}$. The proof is straightforward but not very illuminating. Are there better explanations?