QGIS » QGIS Application

Not all buttons in the advance editing toolbar are available for
all geometry types: you cannot add parts to multipoint or multilinestring feature, but can with multipolygons.
Tested with r 15754 and postgis as a backend:

• You cannot add a point to a multipoint feature (you can only add a new feature). You can move or delete one of its parts . You can copy a multipoint feature and paste it as a new multipoint feature.

• You cannot add a part to a multiline string (you can only add a new feature). You can move or delete one of its parts. You can copy a
  multiline feature and paste it as a new multiline feature.

• You can add and delete parts to multipolygon. (As a minor note, to
  delete a part of a multipolygon. you have to click on one of its
  vertice, which might be a problem if you have a set of contiguous
  (parts of) polygons).

As a side point, note that it is the responsibility of the backend to generate new id's when copy/pasting to avoid duplicates, e.g. with:
ALTER TABLE test_multipoints
ADD CONSTRAINT test_multipoints_pkey PRIMARY KEY;
otherwise duplicated id's are created (which does not prevent most
functions to work apparently).

I consider this as an annoyance more than a major bug because you can always create more features and duplicate the attributes, potentially with a field like relation_id (the OSM way from a conceptual point of view). If two points belong to the same relation (should be a multipoint), then they have the same relation_id. Still, if you already have a database with multi* geometries and must keep this structure, fixing this bug will help.