Voluntary service if your camera is having issues.  Sample size is small, but Nikon is rolling it out for anyone that wants one.  So if you have an older D750, why not get a shutter replacement and upgrade?
@zack3g

I can't find it at the moment, but the shutters aren't a Nikon manufactured part, and it looks like that exact shutter was only used in the D600 and D750.  So that would explain why it's fairly isolated of an issue that's taken so long to diagnose.   Personally, I don't understand why they don't slap the shutter module from the D7000/D800 series into it, as they work flawlessly as far as I've seen.  The 1/8000 shutters have no issues, but the 1/4000 seem to be iffy.

http://www.nikon.com.hk/en_HK/service/service_advisory/d750-shutter-issue

NOTICE TO USERS OF THE NIKON D750 DIGITAL SLR CAMERA OF NEWLY ADDED AFFECTED PRODUCT SERIAL NUMBERS (UPDATED)Thank you for choosing Nikon for your photographic needs
In July 2015 and with an additional notice in February 2016, we announced that the shutter in some Nikon D750 digital SLR cameras manufactured between October 2014 and June 2015 does not function normally, sometimes resulting in shading of a portion of images. Since that time, further examination of the issue has found that the same issue may also occur with D750 cameras manufactured from July to September 2014 and from July 2015 to September 2016.
Therefore, any D750 manufactured from July 2014 to September 2016 is eligible for inspection and repair of its shutter. Please follow the instructions in the “Identifying affected products” section below to check if your camera is affected.
We sincerely apologize for any inconvenience this issue may have caused.
Please direct inquiries regarding this issue to your nearest Nikon service center.

Mine is affected. I just printed a UPS label to send it to the service center.

Nikon advisory

Hooray I received my rebuilt D750 today. New shutter, new front body, new autofocus unit, new rubber grip, adjusted shutter, adjusted autofocus, cleaned sensor and miscellaneous other things. It's like a new camera. Zero cost to me.

It was delivered 1 hour after the solar eclipse ended.

Sucks you missed the eclipse, but basically a new camera is a win.  Sounds like they gave it a full once over instead of just popping in a shutter module.