Skylights that were not installed properly will certainly eventually leak, either directly into the room where it is located; or the leak from a skylight can run along the inside of the ceiling and eventually show itself in another room.
Also, if there has been a re-roofing job done around a skylight it can cause the skylight area to be a problem if appropriate care was not taken. The proper skylight flashings need to be integrated into the shingle roof correctly, or it will cause roof leaks.
Old Roof Usually Equals Old Skylights
Often when an old roof is due for a change it's likely that the skylights are as old as the roof. This could mean that they also need to be changed and the flashings re-done. Even if the skylight has the possibility of a year or two left in it, it may be wiser to have it replaced with the new roof, and done correctly ... rather than find out on some wet November night ...