No, you need more than that, for example, if $X_i$ are iid rvs taking values $\pm 1$ with prob 1/2 each and $Y_{2i} = X_i, Y_{2i+1} = -X_i$. If you avoid next door neighbors these are independent, so they have covariance that declines exponentially, but the partial sums are $\pm 1$

No, you need more than that, for example, if $X_i$ are iid rvs taking values $\pm 1$ with prob 1/2 each and $Y_{2i} = X_i, Y_{2i+1} = -X_i$. If you avoid next door neighbors these are independent, so they have covariance that declines exponentially, but the partial sums are $\pm 1$. You may want to score this as convergence to normal with st. dev. 0, but you can modify it so that the variance doesn't converge by doing as I've sketched for a time, then just take i.i.d's until you get close to normal, then let it let it get back to 0 etc.