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In some simple cases such using the default Monkey as the object to project, the resulting cut looks nothing like the monkey. Note the selected cube only has only been cut around the monkey's eyes, not the entire object.
knife project with the default monkey
Trying with other primitives results in this error

No Other selected objects found to use for projection

I know that I have both objects selected.

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    $\begingroup$ Suzanne is non-manifold, so it just gets the eyes. Knife project is designed to work with flat 2D meshes as the projection object, so I think the overlapping faces confuses it. Collapsing a sphere into a flat circle and removing doubles makes it work as a projection object. $\endgroup$
    – gandalf3
    Commented Sep 20, 2014 at 0:01

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This is because knife project only uses wire or boundary edges:

Knife projection is a non-interactive tool where you can use objects to cookie-cut into the mesh rather than hand drawing the line.

This works by using the outlines of other selected objects in edit-mode to cut into the mesh, resulting geometry inside the cutters outline will be selected.

Outlines can be wire or boundary edges.

To use Knife Project, in 'object' mode select the "cutting object" first then shift select the "object to be cut". Now tab into edit mode and press "knife project".

emphasis mine

A boundary edge is an edge with nothing connected on one side, e.g.:

enter image description here

The primitives, being manifold objects, do not have wire or boundary edges.

Suzanne partially works because it is non-manifold (the eyes are disconnected). The two circles that you get by projecting Suzanne are the boundary edges around the eyes.

enter image description here

The error message is a little bit strange though.

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    $\begingroup$ If Blender documentation could be as clear as your answer... $\endgroup$
    – mins
    Commented Jun 25, 2018 at 18:27
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    $\begingroup$ Inspired by this answer to contribute to the Blender docs.. see changes here $\endgroup$
    – JJones
    Commented Dec 11, 2020 at 4:44
  • $\begingroup$ @JJones Cheers! BTW, barring a bit of Z-fighting, projection of edges internal to the silhouette can work if you Delete > Faces Only imgur.com/a/v4wIO5k $\endgroup$
    – Robin Betts ♦
    Commented Dec 11, 2020 at 11:07
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    $\begingroup$ @RobinBetts Hmm... maybe instead of giving that error knife-project should use the face-less version of the object? Would the "z-fighting" complicate that? Better left as-is? $\endgroup$
    – JJones
    Commented Dec 18, 2020 at 11:54
  • $\begingroup$ @JJones yup.. just twigged gandalf was on it, anyway. $\endgroup$
    – Robin Betts ♦
    Commented Dec 18, 2020 at 12:50

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