Here was my issue in an LTE market. It was working great. Then, even though it had 5 bars of LTE it stopped working (i.e., my ability to use data, even though the phone continued to show a strong 5 bar connection and with the LTE indicator). Also, when I went out of the city to a "4G" market with 4G on the indicator it still could not access the Internet. Then, later in the day back in the LTE area, the indicator completely died. The phone could access the internet on WIFI fine, but not on LTE or 4G. I was on the phone with the special tech support area for LTE and everything was set correctly, data was on, WIFI was off, etc. Finally, we did a complete reset of the phone and the data is working again.
The tech support person said I was the 2nd Lumia owner today with exactly the same problem and the only thing that worked to fix it was a reset. He was going to call the first Lumia owner back later to see if theirs was continuing to work after the reset or if the problems began to recur. He did confirm, by the way, that LTE and 4G was not down where I was so it appears to perhaps be either a software problem with the phone and/or a radio issue.