Unless it's a modem, try hooking it directly to your computer and see if it helps. Sounds like it's a copper cable connection, so it could possibly be the wiring to or around your house, that makes it unstable when too much pressure is put on to it. Might sound stupid, but just to be safe, try with another computer.
Check the first thing first though.
Usually companies like BT have 2 sockets for their phonelines, yet one is a backup. Checking if the internet increases in speed if you use the backup then that means time for a wiring replacement for the main socket.