I was having a lot of issues with my Verizon Note 2 for a while. No matter which ROM I flashed, the best connection was a less than decent 3G connection. After a lot of fooling around and looking for a fix though, I finally figured it out and now I want to share in case other people are having the same problem.
If you already checked to see that your SIM card works in another 4g phone and vice versa and that didn't tell you anything, try this:
-Go to your dialer
-in the menu that it brings up, select phone information
-change the preferred network type to the one that applies to you (for me, it is LTE/CDMA auto (PRL) )
-make sure all the options under that network type selector are enabled (it will say turn off radio, turn off IMS, etc on the button)
-toggle the DNS check so it says "allowed"
After you did all this, you should have no connection at all. No bars. Nothing. Reboot your phone at this point and you should have your 4G!!