The general answer is *Yes*, but not always, depending on what you mean but "cut."
For the *Yes*, here is an example that Erik Demaine, Marty Demaine, Anna Lubiw, and I worked out carefully:

The Latin-cross unfolding of the cube can refold
into precisely 23 distinct convex polyhedra,
as displayed below (all of the same surface area): 
the cube,
two doubly covered flat quadrilaterals,
seven tetrahedra,
three pentahedra, each with one or more quadrilateral faces,
four hexahedra,
and six octahedra:
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![LatinCrossRefoldings][1]
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<sup>Fig. 25.30, p.408 in [Geometric Folding Algorithms: Linkages, Origami, Polyhedra](http://gfalop.org/), 2007.</sup>
<hr />
Here is a "movie" of one of the refoldings to a tetrahedron:
<hr />
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![Latin2Tetra][2]
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But more generally, in the 2012 paper [Refold Rigidity of Convex Polyhedra](http://cs.smith.edu/~orourke/papers.php#303), we showed that *every* convex
polyhedron can be cut open and refolded to an incongruent convex polyhedron.

But if the cutting is restricted to follow edges of the convex polyhedron
("edge unfoldings"),
then there are "refold-rigid" polyhedra.
For example, each
of the 43,380 edge unfoldings of a dodecahedron may only fold back to
the dodecahedron.


  [1]: https://i.sstatic.net/KPyrE.jpg
  [2]: https://i.sstatic.net/r1rTP.jpg