Here's my anecdotal response.
I've used Anker batteries twice in my Note 3.
The first one started to fail, and was clearly bulging after about a year, so I disposed of it at a location allowed for battery disposal because I could literally see the OEM samsung case bulging from it expanding. It was never dropped, or exposed to extreme temperatures, and always in the samsung window case, or the otterbox defender (which may have increased the heat level without me knowing, but that's just speculative so take that with many grains of salt).
Having used Anker batteries many times before, I assumed the first replacement could have been a knockoff, or just from a bad batch, so I replaced it with another Anker of the type. Both were purchased on Amazon. I have had absolutely no issues with the second one, though I do feel the phone get warm when updating my ROM, etc, but that's to be expected. Aside from that the battery life has been very good, with a year and a half of use on it, and I never get concerned about daily use, out side of plugging it into a charger when doing gps/music for a long distance. Even without being plugged in, it stays pretty solid.
Full disclosure for my usage: I'm a light user during the day, aside from calendar, weather, 2 email addresses set to push, and 2 email addresses set to 15 minutes, a few texts, dropbox, drive, onenote, and google all on manual sync, and few phone calls. I'm also on lineage 7.1, rooted, xposed, and bloatware removed or blocked.
TLDR, my current Anker replacement works very well for my needs, but the first one did begin to fail to the point of it looking like it might go into a full failure at a dangerous level, so I can't be sure if it was a knockoff, a bad batch, or just bad luck.