|You do of course realize that daisy chaining switches is a bad idea right?|
Also, it's always best to use a crossover when connecting like devices. I know most, if not all, modern switches are MDIX capable, but, using crossover cables rules out one issue should any problems arise and you find yourself troubleshooting. It's a "best practice" thing one should always keep in mind.
The problem occurs when I connect any computers to the 2960's ports... the first port linking back to the 3560 errors out every time and disables.
I'd wager you're trying to daisy chain that 2960 to #3 switch on the present chain. Stop it! What you really need to do is cascade the switches together. If you don't have a backbone and can't connect the switches singly you really REALLY need to cascade them. This would in effect convert 3 separate (but daisy chained) switches into a single unit (ie: one big switch). This is a lot more effective in regard to resource allocation and bandwidth usage, and, it eliminates the bandwidth overload a daisy chain creates.
This sounds almost like a loop. Are you using STP and/or CDP? Do you have anything else plugged in to the new switch or just a single daisy chain link?
The way I see it, the issue is one of three things:
1) Hardware issue
2) Configuration issue
3) Too many switches daisy chained together
I guess to be able to help you we'll need to have a look at the configuration on the working switches and compare it to the one that isn't working.