According to Nikon, “It has come to our attention that the shutter in some Nikon D750 digital SLR cameras manufactured in October and November of 2014 does not function normally, sometimes resulting in shading of a portion of images. We have decided to take the following measures regarding this issue. We sincerely apologize for any inconvenience this issue may have caused.”
Thankfully it seems that not camera unit has been affected by this. Like Nikon said, this seems to mostly affect those who purchased their camera late last year, although there is a chance that your newly bought model could be one of the affected units if your retailer is still holding onto stock from 2014.
If you’d like to check to see if your Nikon D750 is affected, all you’d have to do is head on over to Nikon’s website, enter the serial number found on the bottom of your camera, and if your unit is affected, you will be able to take it to a Nikon service center who will then be able to examine and hopefully repair the issue for your free of charge, even if your warranty has expired.