Problems tethering my iPad 2 to my iPhone 5

I'm not sure if this is an iPhone problem or an iPad problem, but here it is:

I have an iPad 2 running iOS 6.1 and a Verizon iPhone 5 running iOS 6.1.

When I turn on the Personal Hotspot on the iPhone, the iPad can connect to it and the Internet works for a short time. Then it just stops working. The connection is still active but all attempts to use the Internet timeout.

I also have a Kindle Fire HD that I connect to the iPhone Hotspot. When the iPad stops working if I turn the Kindle Fire HD on and load a web page, the Kindle Fire HD works fine, and then the iPad starts working again too.

I tried resetting the iPad's network settings, but it didn't help.

Help?