I just upgraded to Mountain Lion. Now I can't share files with the Snow Leopard on my network
Some troubleshooting steps to take:


Open the Sharing Preferences on the Snow Leopard Mac, and make sure that File Sharing is checked. Then click Options…* and make sure Share Files and Folders using AFP is checked.
On the Mountain Lion Mac, open Finder and click Go → Connect to Server… (or press ⌘K). In the server address box, enter afp:// followed by the IP address of the Snow Leopard Mac, then click Connect.

If you don't know the IP, you can find it by using Network Utility on the Snow Leopard Mac.

If that still doesn't work, open Terminal on the Mountain Lion Mac, and enter ping followed by a space and the IP of the Snow Leopard Mac.


If successful, you should see output like this:

64 bytes from 192.168.1.10: icmp_seq=0 ttl=64 time=0.249 ms
64 bytes from 192.168.1.10: icmp_seq=1 ttl=64 time=0.207 ms


In which case, there's something wrong with the file sharing set up.
If it's not successful, you'll likely get Request timeout errors, and it's a network issue.



Try those steps and post the results, and I'll update my answer with some more suggestions.