I don't think so. The internet forums are full of desperate people who have fallen into this update trap (Android 5.0 to 5.1) from Google.
I bought the device from digitec in mid-April 2016, set up Wi-Fi, was offered the update to Android 5.1 and carried it out. After that, it was unusable (endless loop between optimising 126 apps and restarting until the battery ran out, even switching it off didn't work).
I then contacted Motorola support and after a few factory reset attempts was allowed to send it to the local repair agent mobiletouch, who passed it on to Motorola. After just under 2 weeks, I had the Nexus 6 with Android 6.0.1 in my hands again.
This is an impressive out-of-box experience from Google, especially since you buy a Nexus device to get Android updates for as long as possible.
There is a well-hidden warning about Android 5.1 under the "Information" section on the Nexus 6 page, which is only displayed when you click on "... more".

1. 5.1 was never faulty?!!!
2. this is a nexus device, the joke of it is, you are always up to date with the latest software including security patches.
3. yes it runs on 6.0.1