Can you establish a physical connection to your modem? Skip the router all together, plug directly into your modem and try it again.
If your Internet stops working under this configuration, we can rule out your router. If it doesn't stop working, then we can certainly blame the router.
I know you said you reboot your router and it works, but this could be coincidental, such as the problem is really with your computer but simply reconnecting to your network fixes the issue temporarily... etc..