Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers , with initial terms . Previous work proved that as the distribution of the number of summands in the Zeckendorf decompositions of , appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in share the same potential summands and hold for more general positive linear recurrence sequences . We generalize these results to subintervals of as for certain sequences. The analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence . As , for almost all the distribution of the number of summands in the generalized Zeckendorf decompositions of integers in the subintervals , appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to , has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and to obtain the result, since the summands are known to have Gaussian behavior in the latter interval.