This article is from the Apple II Csa2 FAQ, by Jeff Hurlburt with numerous contributions by others.
I assume you are using System 7.x File Sharing on the server? If so, you've just discovered one of its inherent limits. Changing computers will make no difference. To be able to have more than ten clients, you will have to run the full AppleShare server software. If you use AppleShare version 3.0, the IIgses can even boot over the network and won't require a local boot disk. If you use AppleShare 4.0 or later, you lose the network boot capability, but in theory the Apple IIgses should still be able to use the server (I've never used anything later than 3.0). You probably cannot run AppleShare 2.x on an LC III, because it only runs under Mac System 6.x. Apart from the number of users, the full AppleShare server adds many useful features, such as administration tools, potentially acting as a print server, and faster performance. It ties up more resources on the machine than File Sharing. AppleShare is commercial, but you might be able to get hold of a cheap copy of version 3 from somewhere. As far as the multiple server option goes: that should work fine. If you have no need to communicate over the network between the servers, it would be a good idea to break the network up into separate segments (server and its block of clients). This will reduce confusion for the users (seeing more than one server), and will improve network performance. In particular, note that LocalTalk is only intended to support a maximum of 32 devices per network segment. If you want 30ish or more computers on the same network, you should be using a router (which physically separates the network segments). By: Rubywand