In order for the net to be able to folded into a polyhedron, there must be an even number of unattached edges! We must also be able to match them up into pairs each of the same length that will join together when the net is folded.
For example, here's net 23. We can see there's an odd number of edges, so it certainly won't be possible to fold it. If the yellow triangle numbered with sides numbered 7 and 8 were removed, it would fulfill our requirement, as the 'new' edge of the red triangle with numbered side 6 could match up with the yellow side numbered 7. However, it's still not certain whether this net would fold up, as the triangles have to be put together to ensure it's possible for the squares to fit in.
There's lots of interesting ideas at http://nrich.maths.org/1381, particularly the fact the angle deficiency of a regular polyhedron (i.e all vertices look the same) add up to $720^{\circ}$. We could use this to predict whether a net will form a polyhedra. Usefully, this also works for concave regular polyhedra like some of the examples in this problem.