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I've got this model of a product:
There are parts of it that are very thin and they don't show up when I slice it in Matter Control like so:

How can I ensure that this comes out as a good print? What's the minimum wall thickness I should have? I'm really new to this and I'm not sure what to do to the model to make it a more successful print.

Your slicer is prolly looking at the extrusion width, and if the wall is thinner than that, it won't print it. Check to see if you can manually overide the extrusion width to a reasonable width and try again. I don't use MatterControl, so someone else may need to chime in as to where to change the settings.

Your wall thickness must be too thin for the nozzle diameter you are trying to use. To check if that is the case change the nozzle diameter in Mattercontrol to be less than the wall thickness.

Or change the model to a larger wall thickness.

I don't know of any other way.

Dale