I had this exact problem with mine. The gap is enough for debris to get in. I could even slide in a business card almost halfway in.
What others have said fixed it for me. Take off the back cover and reapply it from the bottom first. Make sure to press down firmly in each spot.
I took it back to the store and they were baffled at first too but then figured it out. If that doesn't work for you, take yours back to the store. Don't live with it like that - you'll get debris into the battery compartment.
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I got pretty concerned about this problem, so I took it to the store where I got it. The clerk took a look at my phone's back cover and popped it in a new phone, and indeed there was a gap at the same place too. Sticking the back cover of the new phone onto mine, there was NO gap at all. The clerk let me keep the new cover.
When I removed the new back cover and inspected it, I noticed that there were two clips on the bottom of the cover to latch onto the bottom of the phone instead of just one. The one I had had the bottom left clip missing. Clearly that must have caused this problem and was a manufacturing defect. Which concerns me that there might be other phones with this exact same problem (and there really must be, based on what you said above).
A side note: this new cover doesn't provide as much of a satisfying snap as the old one did, although trying to remove it proves it to have been a snug fit. That leads me to a pretty stupid question that should be bleedingly obvious: Are all Note 4 covers made to fit in any Note 4?
Anyway, I am once again a happy owner of a Galaxy Note 4. If I have any advice to give to any of you out there are looking to purchase a Note 4, it would be these two (and I'll post this with many more details on a new thread for anyone to read):
1. PLEASE make sure to visually inspect ALL PARTS of the phone BEFORE leaving the store.
2. Also run some tests of the phone's sensors and radios, again BEFORE leaving the store. A helpful dialer code to help with that:
*#0*#. That'll help you detect any dead pixels, non-working speakers or sensors, and anything else in the phone that could be defective. To test the phone's radios, just run a speed test of a Wi-Fi or 4G connection at testmy.net.