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One possibility is supports. If you had supports "on" then the hinges would have filled with plastic and fused together, which then would cause them to break when you tried to fold it. So first thing: make sure that you have supports "off". Having rafts "on" (that's the mesh) is okay; the thing should print with or without the raft.

Second possibility is that you were printing at a different layer height than what the thing was designed for. Make sure you are using 0.3mm layer height. It can sometimes work at different layer heights but this particular thing was designed for 0.3mm.

Or, it could be that the updated file at http://www.thingiverse.com/thing:440961 might work better. I don't think that will be the case but it's worth a shot if the first two things don't work (but even with that file, make sure supports are off and layer height is 0.3mm).

ORRRRR try using the Customizer to change the hinge clearance a bit, maybe increase it. Some printers will need different clearances than others.

Customizable Hinged Polyhedra