Let me add one last little bit of info: I had trouble with the HIH3160 in the past on RFduino’s I2C library too. The reason was that HIH3160 is triggered to initiate measurement by an empty transmission, but the I2C library was simply skipping further logic if the output buffer was empty at the time endTransmission() is called. I fixed it there by sending one random byte, which got around the issue.
I didn’t originally suspect I2C library issues because someone else had posted an HIH61xx library, making me suspect it was working fine. However, I’m out of other explanations so I thought I’d put it out there. Perhaps the slightly unusual I2C protocol of the HIH6130 is conflicting with the I2C library here?