I'm on AT&T and most of the time my lte on my iPhone 5 won't work. I'll see glimpses of it but it will stay on 4g most of the time. My area is covered with LTE
Depending on where you are, it's normal behavior. Covered doesn't mean 100% coverage. My iPad 3/iPhone 5 have coverage probably half the time. I'm in Los Angeles and Los Angeles is supposed to be covered.