Wanted to make a quick post and let you all know.
If you are like me, and tired of waiting for device updates, on a network that is at least for me getting progressively WORSE (5mbps download on 4g LTE. FUN!) that you are not stuck.
You can indeed put a tmobile sim into your note 3.
You can enter some APN settings and get what tmobile calls HSPA+ speeds on your device.
Additionally, you can enter a service mode that allows the 4th band that tmobile uses for their LTE to work on the phone, so long as you are in a larger market and have that available via tmobile.
Now, VZW will tell you on the normal customer service line this is not possible.
This does not hack your phone
This does not mod your warranty.
Even just putting a tmobile sim in the phone and dialing works so long as the GSM mode is active.
THE DEVICE IS A WORLD PHONE, UNLOCKED AND THIS COMES FROM VERIZON'S OWN ADVANCED TECH SUPPORT.