In most cemeteries that are still in use, when they are ‘full’ they will simply be closed to new burials, maintained, and a new cemetery will be opened (usually outside of town/city limits due to space constraints as a result of development). Occassionally there will still be burials in the closed cemeteries, for families that have reserved plots, etc. It is becoming more common for cemeteries to be combined for numerous parishes or municipalities, simply because of space and cost.
However, in history, many cemeteries were used continuously, either being expanded throughout time (where space allowed) or by exhuming individuals to create space for new burials. This was quite popular pre-Reformation and led to the creation of charnel deposits and ossuaries (of which many were destroyed during the Reformation). Exhumed individuals could be placed in charnel deposits, ossuaries, re-buried alongside a new burial, etc… often they were already skeletonised, other times flesh was removed to make them easier to handle.