Can a Verizon iPhone 5s use T-Mobile LTE?

I just purchased a Verizon iPhone 5S gold and am running it on T-Mobile. I get 4G and LTE wherever it is available.

Short answer, yes (assuming you have LTE coverage in your area by T-Mobile). But you should get the iPhone 5s from Verizon instead of getting it from T-Mobile. Also, carrier unlocking has nothing to do with 3G or LTE.

Long answer follows below.

The iPhone 5s models A1533 (GSM) and A1533 (CDMA) support the LTE bands that both Verizon and T-Mobile use.

If you look at the technical specifications for these two iPhone 5s models (A1533 GSM and A1533 CDMA), you'd notice that:

• The LTE band coverage is the same across these two models

• The GSM model supports only GSM (so you cannot switch to Verizon even if you want to, since Verizon is a CDMA carrier)
  [Model A1533 (GSM): UMTS/HSPA+/DC-HSDPA (850, 900, 1700/2100, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz); LTE (Bands 1, 2, 3, 4, 5, 8, 13, 17, 19, 20, 25)]

• The CDMA model supports both CDMA and GSM
  [Model A1533 (CDMA): CDMA EV-DO Rev. A and Rev. B (800, 1700/2100, 1900, 2100 MHz); UMTS/HSPA+/DC-HSDPA (850, 900, 1700/2100, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz); LTE (Bands 1, 2, 3, 4, 5, 8, 13, 17, 19, 20, 25)]

So you would be better off going for the iPhone 5s CDMA [Model A1533 (CDMA)] on Verizon and then move to T-Mobile.

Also see: The Verizon iPhone 5s comes unlocked just like the Verizon iPhone 5

I have the Verizon 5S and I swap my T-mobile sim and Verizon sim with no issues. I get all T-mobile bands. I am grandfathered into Verizon Unlimmited data so I have to buy all my phones full retail. Recently I decided to give T-Mobile a test run to see if it would work for me and save me a boat load of cash every month. So far I have had no issue on any T-mobile bands Edge,HSPA,LTE.