The Possible Winner (PW) problem, a fundamental algorithmic problem in computational social choice, concerns elections where voters express only partial preferences between candidates. Via a sequence of investigations, a complete classification of the complexity of the PW problem was established for all pure positional scoring rules: the PW problem is in P for the plurality and veto rules, and NP-complete for all other such rules. More recently, the PW problem was studied on classes of restricted partial orders that arise in natural settings, such as partitioned partial orders and truncated partial orders; in particular, it was shown that there are rules for which the PW problem drops from NP-complete to P on such restricted partial orders. Here, we investigate the PW problem on partial chains, i.e., partial orders that are a total order on a subset of their domains. Such orders arise naturally in a variety of settings, including rankings of movies or restaurants. We classify the complexity of the PW problem on partial chains by establishing that, perhaps surprisingly, this restriction does not change the complexity of the problem, namely, the PW problem is NP-complete for all pure positional scoring rules other than the plurality and veto rules. As a byproduct, we obtain a new and more principled proof of the complexity of the PW problem on arbitrary partial orders.