I've had this issue on several Amazon devices. The issue as best as I've been able to diagnose it has to do with how Amazon devices handle DHCP. Traditionally a device get issues an IP address for a certain amount of time, we'll say 10 minutes for simplicity. At half of the lease the device will renegotiate the IP address lease. So at 5 minutes the device will renew its IP address for another 10 minutes. That's the traditional way.

Unfortunately the way Amazon does it is that it waits until the last second to renegotiate the lease. While these leases happen in a few seconds it will often result in the devices dropping connection.

TLDR. Assign a static IP in your router for your Kindle Fire HD.