I don't think so, because the same problem occurred with my girlfriend's iPhone 5S and she never uses any Bluetooth device.
The only solution we have to figure out here in Hungary is to call our carrier and ask it to disable the "hide Caller ID" function. There are phones with three option for "send caller ID" function. Send, hide and "use carrier settings". If you select the "use carrier settings" option, the phone will use the the carrier's default settings. For example the Hungarian Telekom usually hides the ID, but the Telenor doesn't. But if the "hide caller ID" function is enabled on the carrier side, BUT you choose "send caller ID" on the iPhone settings, the iPhone's setup should override the carrier's settings. Due to the lack of "use carrier settings" option the iPhone should always override the carrier's setting in that way you choose, but it looks like this function is not working properly.
So the best you can do is call your carrier to disable the "hide caller ID" (or enable the "send caller ID" depending of the carrier) function on your number. It worth a try. (By the way you can try to use your carrier's app (if it exists) to set it up correctly)