Just a suggestion here...
If you have multiple access points set up with the same SSID from different manufacturers, go beyond just checking that the SSID and password to access the network is the same...also go a bit deeper and check they actual type of encryption used. For example, I put a new router in the house to cover a weak signal area configured just as just an AP. The SSID and password to access the network was a match for my primary router. However, there was a subtle difference I missed in the initial configuration, where the primary router was set to WPA2, and the secondary AP was set to WPA/WPA2. This had the unfortunate effect of using only AES on the primary router, but TKIP on the secondary AP. Wireless devices would not shift seamlessly between the two access points with this minor configuration difference, as they had to reauthenticate with a different mechanism each time. My Galaxy Note 3 would shift back and forth all the time, reporting back an unstable network configuration.
This was easy to clean up, I just set everything to WPA2/AES only. I also realized I had no more 802.11g devices in the house, and set both routers/AP's to support 802.11n only.
With a Galaxy Note 3, you can "scan" the details between multiple access points as you move around by dialing in *#0011# Choose the wifi menu from the menu button. A high level of detail on the wifi connection will be revealed. Move between the AP's that the phone thinks are "unstable" and see what settings change. If there is a difference in the encryption used as you move away from one and closer to another, you have a configuration issue where the AP's don't have critical settings set up as a match.
This could also happen if you authenticated to two entirely different networks in close proximity, like a neighbors open network either deliberately or by accident. Your device may jump back and forth between the two. Make sure you force the phone to "forget" the network that is not yours to avoid this.
Minor setting differences can be easy to miss, especially if the configuration screens are different due to the two pieces of equipment being built by different manufacturers, or entirely different networks in close proximity. Best if you make sure its all set to WPA2/AES only, and generally always avoid connecting to "open" networks. Force the "forgetting" of any open networks you previously authenticated to, to avoid the chance your phone will lock onto one just because someone used the same SSID of a previous open network you came across.