If the walls are plastered brick, the cabinets can easily be rehung. One method is to drill new holes two inches below (or above) the old ones. The other is to clean out and reinforce the holes (thus is good practice anyway in walls that are crumbly, which lime mortar often is)
You remove the old plasplug. This will pull out some of the crumbly old brick or mortar. You clean out remaining dust using the hose of a vacuum cleaner. You put the nozzle of a tube* of builders adhesive of the "no more nails" variety deep into the hole, and inject it, starting at the back side there is no air bubble, until it is almost full. You then press a long plasplug into the hole, using a screw a couple of turns into the plug as a handle, pressing it below the surface of the plaster and smoothing off the adhesive. You leave it to set firm (will take a day or so). When set, you remove the "handle" screw. You will hang the cabinet on a long screw, which passes through the plaster (which has no strength) and at least an inch into the brick beneath. Although the adhesive is not very strong, the screw will cause the plasplug to expand. Since the hole is completely full of adhesive, the plaster cannot expand, or crack or break, it simply transmits the expansion force to the brick in which it is deeply embedded. Thus there must be no air gap. The plug and the screw need to penetrate the wall by at least two inches. To position the screws accurately, measure the horizontal and vertical positions in advance, and rule pencil lines, leaving a cross at each screw position, which will be visible after the drilling and filling. Decorators filler or plaster are totally unsuitable.
- builders adhesive comes in large cylinder, and is applied using the same type of "gun" as silicone sealant round a bath, or decorator's caulk.
There is a similar, but better and more expensive, method for outdoor or structural fixings, injecting a plastic resin into the hole
There is a different method for cabinets that have an adjustable hanging hook inside, that hangs onto a metal strip screwed to the wall.