The Possible-Winner problem asks, given an election where the voters' preferences over the set of candidates is partially specified, whether a distinguished candidate can become a winner. In this work, we consider the computational complexity of Possible-Winner under the assumption that the voter preferences are $partitioned$. That is, we assume that every voter provides a complete order over sets of incomparable candidates (e.g., candidates are ranked by their level of education). We consider elections with partitioned profiles over positional scoring rules, with an unbounded number of candidates, and unweighted voters. Our first result is a polynomial time algorithm for voting rules with $2$ distinct values, which include the well-known $k$-approval voting rule. We then go on to prove NP-hardness for a class of rules that contain all voting rules that produce scoring vectors with at least $4$ distinct values.