I don't think it's normal if it happens all the time. For example, I'm on 4G with a full signal right now. There are occasions when I'm in a 3G area but I don't typically get full signal.
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Doesn't make a whole lot of sense unless for whatever reason VZW decided to put a LTE tower near your work where there wasn't a 3G tower. Again, doesn't make sense because they often just add LTE to 3G towers. Generally speaking, you can have a strong 3G signal and a weak LTE signal, but if you have a strong LTE signal you will have a strong 3G signal.
YES! I often find myself with full bars of 4G but never full bars of 3g... If anything it's 0, 1 or 2... Usually when I'm not on 4G the internet speed is unbearable. I think I have bad radios.