Partial Waiting List is - in a PNR/ticket, 5 passengers are travelling, out of which tickets for 3 persons are confirmed and 2 passengers are in WL (nos. are indicative only, it may be x, y,z) then it is known as Partial Waiting List.
In this Partial WL case, in case the tickets are booked throu irctc (E-tickets), the WL tickets are cancelled automatically and refunded back to the customer. In case of I tickets (issued by Res. counter/IRCTC also - original printed tickets of Rlys) if the WL passenger wishes to cancel their journey, can do so. else, there is an option of adjusting with the confirmed berth passengers - but this entirely depends upon the TTE of...
more... the train whether to allow you to board or not. Discretion of TTE. You can request and adjust with other confirmed pass. You may have to pay sleeper charges, but will not be allotted any berth. REceipt will be given. - In case of no show passenger, TTE may allot berth if available.
Of course, during this summer rush, it does not happen, but there may be chances.