The oddest problem

Not sure if this is the right place to post this but here goes.

I have a wired network where I work. Last week I had my ISP check my internet speeds. They changed to a new modem that has a router built into it. Without getting into a long story the new modem and a different IP address than the old one. Apparently this caused all kinds of havoc and not only could my computers not log into the web I could no longer log into my file server. After another tech came by a few days later he determine that since the new modem had a different IP address it could not communicate with any of my devices. He then changed the address to what the old modem had and bingo my file server came back online. All is good right? The speed tests seemed much better. The next day I found web pages loading slow or not at all. Then all of a sudden they'd load. Another call for a tech and the third different one came by. He saw the erratic behavior of the connection too. He changed out the line and a splitter outside and thought all was good. But Noooooooo. Web pages would not load or would load then hang up when trying another page.

The odd thing is I logged in to a music stream and it never shut off even though I could not access the web. Of course it would access pages once in a while. I checked the network diagnostics on my Macs and they would say the connection was good. When I noticed I could not access a web page I would check diagnostics again and it would tell me to restart the device meaning the modem I assume. The lights on the modem were all good like it was working properly. The techs tried to blame my network switch saying it may have a bad connection. I found that improbable but thought I'd do a simple test. I unplugged the ethernet from my computer and activated Airport. It showed a good strong signal coming from the wi-fi on the new modem. I could access a web page then it would shut down just like before when it was still wired into my network. So this eliminated any of my equipment from the equation.

There are several things I don't understand. First, why would the connection be on then off and seem to get worse over time. Second, if music keeps streaming the whole time but I can't access a web page what does that mean. Third, what the **** do I do now? I'm no expert but I never had an issue before until the new modem was installed. To me this has to be the cause but doesn't make sense since the modem appears to be working properly and the techs felt everything was set up as it should be. Anyone have a clue as to what I might be missing?