Insulation helps, but "cold bridges" can get you. Cold spots in insulation get locally colder than in uninsulated walls (because that little uninsulated bit gets to leak heat to a larger area of the outside than it did before: it's no longer competing against the insulated bits to dump its heat) and if that brings the uninsulated spot below the new dew point (indoor air is warmer after insulation so may be carrying more moisture) you get localised problems where there weren't any problems before. Similarly inside corners get a lot of outside surface opposite them and the junctions are the hardest bits to insulate.
In our previous flat, a cross-wall 1960s building with a single-block skin under wooden boards,and a barely-insulated solid concrete roof, I spent a lot of the winters bleaching window frames and the corners of the ceiling. It was plenty draughty, too, especially after we removed most of the badly-conceived secondary glazing which was trapping moisture between itself and the windows, though that didn't really help. We just ran dehumidifiers constantly during the winter. The years with washable nappies were particularly bad.
And then we moved out to a house we could insulate and ventilate properly. It's so nice.
