Thanks everyone!
In my case it probably was that the model was too high poly . I tried importing it into Meshlab and it crashed Meshlab. It's a model I created in OnShape and exported in fine resolution. I exported in medium and coarse and they also crashed Meshlab! However a very similar fine resolution model has been successfully CNC milled by Protolabs
I then slightly simplified my model (removed some fillets), exported it at medium and loaded it into Cura. I changed the layer thickness from 0.1 to 0.15. It's now almost finished printing without a problem. I'm not sure which of the changes solved the problem.
I'm surprised even a coarse export of my model (as STL) would not import into Meshlab, considering that Meshlab is for simplifying and repairing models. My model is not really complex compared to some I've seen although it does have a few sub-mm features.