There's a partial clog somewhere in the line. It is letting some flow by, but slowly. It could be numerous things - roots, a kink in the line, a collapsed line (especially on an old house, it could be tile drain lines which are more prone to letting roots in or cracking and collapsing).
A drain cleaning attempt all the way to the main line in the street seems called for. If they can't get a snake through, or find a lot of roots, it may be time for a new line.