Bad GFCI or did I wire it wrong?
Always pause in your installation after you have fitted the LINE wires, but before you have fitted any LOAD wires. Power up and run the GFCI through its paces.  

This clears a number of problems, including 


getting supply and downline wires confused
accidentally attaching to the LOAD terminals
shorts and other problems with the downline wiring
having an actual ground fault somewhere downline


If the GFCI works without anything on the LOAD line, then you know that if problems emerge after adding a set of wires to LOAD, this is a problem there, and can troubleshoot accordingly. 

Another GFCI device on a different circuit cannot cause this.  Another GFCI upstream on this circuit would cause this GFCI to go dark entirely.