Unfortunately I'm having this very same issue with my iPhone 5 without success from any methods listed here. I've tried all the software settings, resetting my network settings, new nano-SIM, and restoring the phone. When I restored the phone, it won't activate through iTunes or through Wi-Fi access. "We're sorry, we are unable to continue with your activation at this time." This is highly frustrating and I've ended up buying a new phone because I can't use the iPhone at all. I even went as far as having a phone repair shop open the phone and try a new antenna - that didn't work either. So now I'm down to the problem being a software issue with Apple.
Back story: I bought the phone (unlocked) from another military member here in Germany last year in November and have been using it ever since then without a single problem with the SIM or antenna reception. On 18 January, I woke up to see my phone had "No Service" as the status. The phone hadn't been dropped, fell in water, or anything else to damage it or cause it not to work. I've had it in the Otterbox case I purchased when I got the phone ever since.
Unless an Apple Store employee can resolve this issue, I will NEVER buy another iPhone again. I owned the original upon release, the 3G, 4 and 4S without ever having a single problem until now.
If anyone else reading this is considering buying a 5, 5S or 5C I suggest you steer clear until they get this problem sorted out. I didn't buy an iPhone to have this many problems.