Hanging connection/no connection

I seem to have a contrary connection problem. My internet will work perfectly happily for a period of time and then suddenly take a long hanging time to connect and eventually not connect at all.
The first time an engineer suspected a faulty modem and sure enough a replacement worked. He left it on approval and all has been okay for several weeks. yesterday the connection started to hang again and this morning again no connection. I tried all the troubleshooting routes with no success and out of desparation reconnected my old modem back again and "hay presto" here I am back again.
I have a feeling this hasn't solved the problem though and would be grateful if anyone can suggest how to cure this problem.