It's probably an overheating problem. During reboot, there is no demand to connect to use the internet so either the NIC in your PC or the router is cooling down a bit during this time, and then you're okay again.
I've seen this problem many times, and it often turned out to be this; the fix is really easy -- vacuum the dust out and make sure the router is in an area that doesn't prevent airflow (e.g., in an area as small as a shoe box prevents airflow, but up on top of a shelf that is open doesn't prevent the needed airflow).
Updating the NIC driver to the newest version sometimes helps to resolve this problem as well, but I'd try clearing any dust first.