Two computers will not access internet at the same time.

I have two computers that I'm trying to hook up and access the internet with at the same time. Each computer has it's own modem. However, both modems are feeding from the same phone cord in this room (if that should matter). Anyway, I can only get one or the other to access the internet, not both at once. Could my ISP (Verizon) somehow have me limited to how many computers I can hook up? Or, could this one phone cord possibly not be enough to carry the load of both of them? Anyone have any ideas? It's always been my understanding that you can hook up multiple (many more than two) computers via this method. I can't figure it out. I've tried everything I can think of Any and all ideas, suggestions, etc., will be most appreciated. dawsontimothy