I had the Galaxy Nexus for a year, from what I read in this forum and in others is thas the Gnex has one of the weakest radios, so connectivity to 3G and LTE was a bit bad that it also affected battery life. I used to carry 2 high capacity batteries.
Also, I know that when we got the 4.3 update, it created a bug when connected to LTE and you would fall back to 3G, data would not work unless you toggle airplaine mode. so maybe this is what you are experiencing ?
After a month of all these issues, decided to upgrade to the Moto X even tho it was not pure stock.
get better LTE and 3G bars where I had minimal service such as my office.