To many things conected to internet?

[Arthur]

Maybe you are maxing out your bandwidth . Thats the reason your unable to access the internet since other applications are eating away the little you have. Restarting the modem cuts the connection with which ever devices are connected making the issue seem to go away. I think you should try to disconnect everything for a day and only connect one device and monitor how the modem/rourter handle it . If it works fine it means you are maxing out on your bandwidth.
If its the case just be more conscious of what is connected at what time and if it really need to be on the Internet.