It may depend on the version of OS X being used but the following worked for me and may describe something that will help you.
As others here have indicated, normally you can only have once user logged in to a file server from a Mac at a time, or at least only one user that works properly. This restriction historically has applied to network login accounts and network home directories rather than a group share or a general file share. Obviously if you can't have two network logins at the same time on the same Mac, the issue of then not being able to access the same group share or general file share is not one you hit because you are blocked before that stage.
This is down to the way network shares are mounted and accessed by accounts. Normally when a network login account is used the first (and only) network login mounts their network home directory on to the client computer in the /Volumes directory. This mount is 'owned' by the user logging in and that specific mount is therefore not accessible by other users on that client Mac. What makes things particularly problematic for network home directories is that Open Directory is set to use a specific path for a user network home directory. Which leads us on to the second issue...
It is hypothetically possible to have two different users logged in on a client Mac, and have each login to the same share. If this is done, the second user cannot use the first users mount point because it is 'owned' by the first user, and the automount system will mount the share under a different path so you might end up with the following
User1 = /Volumes/FileServer
User2 = /Volumes/FileServer-1
even though they are both accessing the same share on the same server which has the same name. As mentioned above in the case of network home directories which have to be accessed via the path defined in Open Directory User2 will fail because their version of the path is different to what Open Directory says, and if they try using the path from User1 they are denied access. However for a general file share the actual path is less important in most cases so theoretically the user could go to /Volumes/FileServer-1 and access files successfully.
The real-world instance I had a similar setup that worked was as follows. I had a Mac XServe running as a Terminal Server using iRAPP. Users connected to the XServe via Microsoft Remote Desktop Client and logged in as network login accounts. I used NFS for the network home directories which avoided the issues above. However these users all still wanted to access a Mac FileServer (a different XServe) via AFP. This worked but as mentioned above each user got their version mounted with a different name. If I had five users logged in then I would have the following mounts
/Volumes/FileServer
/Volumes/FileServer-1
/Volumes/FileServer-2
/Volumes/FileServer-3
/Volumes/FileServer-4
Each of these showed up on each users desktop (via Microsoft Remote Desktop) and they could only access in the Finder their own. If I was using an Application that was using a specific file path it would potentially get confused as to which version to use, in my case this was an AppleScript but I worked round it by testing (in a loop) which version allowed access to a file without error and then knew that path was the correct one.
In conclusion you might be able to have multiple users access the same share via AFP from the same server but may find that you need to access it in a slightly different way.