I have a Verizon Note 3 and I came back from the Philippines two weeks ago. I bought the SMART sim card from there and all I got to work was voice and sms. When I looked at the Mobile network type under the status menu, I think it read EDGE or HSPA+. I spent 5 days in the province so finding an internet connected computer to look up how to get my phone on the internet was not an easy task.
Unfortunately, I wasn't able to find out how to get the correct APN settings before I left. My dad has the Verizon iPhone 5 and I put a SMART sim card in there as well. Same result, voice and sms only, no mms. But hey, the Note 3 works overseas with a foreign sim card!