What is the actual version of Lightroom you are using? Anything over 5.7 should work, LR5 is at 5.7.1.
ACR 9 should be fine as long as you have a newer version of PS that can use ACR 9.
Are you opening LR 6 or LR 4? Use Help / About Lightroom to find out.
If actually LR 6 then the D750 files may be corrupted by viewing or copying them with an old version of Nikon software.
Try copy the NEFs with the OS function like Finder or Explorer then see if they import.
There is a utility to fix the corruption that obsolete Nikon software causes:
Thanks ssprengel, it was the old nikon software. With the new Nikon Transfer 2 all is fixed. Thanks for the insight.
I installed CameraRaw 9.0 as an option, but that does not seem to have done the trick either?
Just for information - ACR has no relevance whatsoever on whether Lr will open a file.
The D750 has been supported since November 2014 (Lr 5.7) so you're either using an older version of Lr, or - depending on what actually happens when Lr "doesn't work" - this might be (as Steve mentions above) down to something like the files having been touched by Nikon software on import (which does something to the file headers that renders them unreadable by Lr); or a problem with the Write permissions on your machine.
So - some actual details of what you're seeing would be helpful, along with your OS.