A partial instance is an instance that is being cached but only part of which is available in the local cache. There are three ways in which partial instances can arise: client applications requesting only part of an instance (Adobe's Acrobat Reader plugin is famous for that), a server dropping a connection mid-transfer (because it is short on resources, or, surprisingly often, because it is buggy), a client dropping a connection (usually because the user pressed stop).
When an instance is requested that is only partially cached, it is possible to request just the missing data by using a feature of HTTP known as a range request. While support for range requests is optional, most servers honour them in case of static data (data that are stored on disk, rather then being generated on the fly e.g. by a CGI script).
Caching partial instances has a number of positive effects. Obviously, it reduces the amount of data transmitted as the available data needn't be fetched again. Because it prevents partial data from being discarded, it makes it reasonable for a proxy to unconditionally abort a download when requested by the user, and therefore reduces network traffic.
Polipo caches arbitrary partial instances in its in-memory cache. It will only store the initial segment of a partial instance (from its beginning up to its first hole) in its on-disk cache, though. In either case, it will attempt to use range requests to fetch the missing data.