I'm not quite familiar w/ chestnut dialer, but I'm gathering you are using a 56k connection?
Now, if you are on dialup, what you want to do if it states it's connected, then first go to the terminal, and run the command "/sbin/ifconfig", and the deviceID should come up with an IP address, so for example, most standard Cat5 connections would be "eth0", and then would show the IP Address (i.e. 192.168.1.101), and some other TCP/IP information.
See if you are getting a valid IP address. If you get the address 169.254.*.* (whatever the last 2 octets may be), that means TCP/IP gave up resolving the IP address.
So, if at that point, you don't get an IP address, or maybe you do, but it's not connecting to the internet still, do the command "/etc/init.d/network stop" and hit Enter, then "etc/init.d/network start", and hit Enter. That should restart all network services, including DHCP, TCP/IP, and DNS.
That command may very slightly if SAM linux doesn't use the System V style init scripts. Try that, if you have no luck, post another reply, and we'll go from there.