I love my Nexus 6, but since day one the charging port has always felt like most USB cables didn't go all the way in. As it approached the 2 year old mark, this condition had gotten steadily worse to the point that any and all cables would simply fall out given a stiff breeze.
I searched and searched online, but as often happens if you don't create the exact right set of search terms, I came up pretty empty. A lot of articles blamed "non-OEM" cables, others said they sent theirs back to Google for a replacement.
Even though it was way out of warranty, I did contact Google. They had no answers, suggesting I contact Motorola for hardware support. I was disappointed, but it triggered an idea- I started searching for articles about loose connections on Motorola phones, and not specifically the Nexus 6. Now things started coming up, and with a little digging I found this post on XDA Developers: http://forum.xda-developers.com/showthread.php?t=1856180
In it, user telmosousa describes how using a toothpick, you can clean out the port on your phone. The toothpicks I had were even still too large, so I literally had to whittle one down to get into the port and... sure enough, a disgusting amount of lint and other crud scraped right out. My charging port is back to its like-new "not quite looking all the way plugged in but nonetheless fully functional" condition!
Whenever I find a "how to" that took me forever to find, I do try and post it here for others!
Hope this helps some people out!