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Zeckendorf's theorem states that every positive integer can be writtenuniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initialterms $F_1 = 1, F_2 = 2$. Previous work proved that as $n \to \infty$ thedistribution of the number of summands in the Zeckendorf decompositions of $m\in [F_n, F_{n+1})$, appropriately normalized, converges to the standardnormal. The proofs crucially used the fact that all integers in $[F_n,F_{n+1})$ share the same potential summands and hold for more general positivelinear recurrence sequences $\{G_n\}$.  We generalize these results to subintervals of $[G_n, G_{n+1})$ as $n \to\infty$ for certain sequences. The analysis is significantly more involved hereas different integers have different sets of potential summands. Explicitly,fix an integer sequence $\alpha(n) \to \infty$. As $n \to \infty$, for almostall $m \in [G_n, G_{n+1})$ the distribution of the number of summands in thegeneralized Zeckendorf decompositions of integers in the subintervals $[m, m +G_{\alpha(n)})$, appropriately normalized, converges to the standard normal.The proof follows by showing that, with probability tending to $1$, $m$ has atleast one appropriately located large gap between indices in its decomposition.We then use a correspondence between this interval and $[0, G_{\alpha(n)})$ toobtain the result, since the summands are known to have Gaussian behavior inthe latter interval.

Gaussian Distribution of the Number of Summands in Generalized Zeckendorf Decompositions in Small Intervals