Gmod messing up entire internet

[Doktor Delta]

I just bought Gmod recently, and for about a week now I've noticed something that happens every time I play online. If I don't immediately join a server, or if I disconnect from any server, my entire internet goes down. The connection still appears as 100%, but I can't access any website or pull any sort of content from the internet from any web-enabled device in my house. After about 10-20 minutes, the internet is fine again. Is this something I can fix?

INFO:
Verizon FIOS
Computer with Gmod is using wireless connection

[Stephen]

I have the same problem. I have Verizon DSL. There are several routers between me and the Verizon DSL modem, but launching a gmod server causes my Verizon DSL modem to lose its connection.

Edit: I should clarify: When I host a server in gmod, it kills my internet connection. I had thought the system plugged into the DSL modem or the DSL modem itself were faulty, but it's only when I launch a gmod server that I lose my connection. The routers and systems between my desktop computer where I'm launching the server (via srcds_run or gmod via steam, both have this problem) are wired. I can still access local network resources and access each of the different subnets in my home, but the DSL modem itself loses its outside connection.